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ESTABLISHED 1845. 


W . & lv, tC. GURLEDY, 'I'roy, N. Y. 

MANUFACTURERS OF 




All Instruments sent to the purchaser Adjuste 
and Ready for Use. Send for full illus¬ 
trated price list and circular. 










































































A MANUAL 


OK 


Land Surveying 


COMPRISING 


AN ELEMENTARY COURSE OF PRACTICE 
WITH INSTRUMENTS 


AND A TREATISE UPON THE 


Survey of Public and Private bands, 


PREPARED 


For use of Schools and Surveyors. 


/ . 

By F. HODGMAN, M. S'., C. E., 

• * 

Practical Surveyor and Engineer. 



“Let things that have to be done be learned by doing them” 


F HODGMAN, 
Climax, Michigan, 
1897. 









Entered according to Act of Congress in the year 1897, 
By F. HODGMAN, 

In the office of the Librarian of Congress, at Washington- 



~TA SSI 

■H 6*0 






PREFACE. 


This addition to the already numerous treatises on land 
surveying was caused by the demand of the surveyors of 
Michigan for a treatise which would deal with the prac¬ 
tical questions which meet the surveyor in his every 
day work in the field. Several admirable treatises were 
already in existence which dealt amply with the mathe¬ 
matical and instrumental part of surveying. But the 
perplexing questions which meet the surveyor are not 
questions of mathematical calculation or of the use of 
instruments. On the contrary they are, for the most part, 
questions of how to apply the principles of common law 
and statutory enactment to the location of boundary 
lines. These are the controlling considerations in all re¬ 
surveys; a class which comprises probably nine-tenths of 
all the land surveys which are made. Scarcely an allusion 
to these principles was to be found in any of the works 
on surveying extant. In 1880 the Michigan Association 
of Surveyors and Civil Engineers appointed a committee 
on manual, to prepare a work which would give authori¬ 
tative answers to the many questions of practice which 
came up before them. The committee spent their spare 
time for five years in an exhaustive research of the laws 
and the decisions of the highest courts in the land. The 
chairman attended the meetings of various surveyors’ 
associations and collected their reports. From the great 
mass of material thus collected, the leading points in the 
laws of the United States and the decisions of the courts 
of last resort were selected, covering, as nearly as possible, 
all the points relative to surveys and boundary lines 
which arise in the land surveyor’s practice. The legal 
decisions quoted are a part of the Common law of the 
whole country and apply wherever the Common law 
prevails, whether in Canada, England, or the United 

[iii] 





IV 


PREFACE. 


States. It should be remembered, however, that differ¬ 
ent courts do not always expound the law alike, and 
sometimes a court reverses its own decisions. When¬ 
ever there appears to be a conflict of authorities, the 
Surveyor should follow the latest decisions in his own 
State if there be any. It seemed to the committee 
to be important that the student in land surveying 
should be taught these things; that they were as 
necessary for the beginner to know as for the older 
practitioner, and hence might.properly be incorporated 
in the text book. Having this in view, it was decided to 
extend the scope of the manual by including such mathe¬ 
matical work as would make it equally adapted to the 
use of the student as a text book and the practical sur¬ 
veyor as a book of reference. In preparing this portion 
of the work, the leading idea has been that, so far as 
possible, the student should be taught by actual practice 
in the field, as well as in the class room; that he should 
learn to survey by surveying. The solution of a problem 
in surveying in actual practice is always worked out 
upon the ground, hence suggestions are made to the 
student how problems may be solved, instead of giving 
any formal solution. It is pre-supposed that every 
successful teacher will have methods of his own for 
conveying instruction, and will use these suggestions or 
make different ones as may seem best to him. Doubtless 
things have been omitted which some would regard 
as important to have introduced. Such omissions will be 
supplied by teachers at their pleasure and convenience. 
We acknowledge our indebtedness to the authors of 
many treatises which have been consulted in the prepara¬ 
tion of this volume, especially to the works of Davies, 
Gillespie, Hawes and Dunn, also to Messrs. W. & L. E. 
Gurley for many favors received, and to the officers and 
members of the Surveyors’ Associations of Michigan, 
Ohio, Indiana, Illinois and Missouri for many valuable 
suggestions, sympathy and assistance. 

F. IIODGMAN. 


Climax , Mich., 1891. 




TABLE OF CONTENTS. 


CHAPTER I. 

DEFINITIONS. 

PAGE. 

INSTRUMENTS FOR MEASURING DISTANCES. 

The Chain_ 2 

The Steel Tape_ 3 

Marking Pins_ 4 

Measuring_ 4 

MEASURES OF LENGTH AND AREA, English_ 9 

Old Spanish_-_ 11 

Old French_ 13 

Standard Measures_ 14 

THE PICKET, To Run Line with_ 1G 

To Pass Obstacles_ 18 


CHAPTER II. 

INSTRUMENTS. 

THE SURVEYOR’S COMPASS, Description of_ 19 

Adjustments of- 22 

Electricity- 24 

To Run Lines with- 24 

To Pass Obstacles- 25 

THE MAGNETIC NEEDLE, Changes in direction of- 2G 

Local Attraction- 27 

Difference in Instruments- 27 

Things to be Observed- 27 

Marking Lines- 28 

How to find a True Meridian- 28 

THE TRANSIT, Description and Adjustments- 34 

How to use- 41 

Assistants and their Duties- 42 

The Color Pole_ 43 

Projecting the Line- 44 


[V] 































VI 


TABLE OF CONTEXTS. 


CHAPTER 111. 

INSTRUMENTS, CONTINUED. 

THE SOLAR COMPASS, Description and Adjustments- 47 

How to use- 55 

SOLAR ATTACHMENT TO TRANSIT. 

Description and Adjustments- 63 

How to use_ 68 


CHAPTER IV. 

MEASUREMENT OF ANGLES. 


TO MEASURE ANGLES, With Tape and Pins- 70 

With the Compass- 73 

With the Transit....— 83 

Verniers_ 85 

TO CORRECT RANDOM LINES, Of one course_ 75 

Of several courses- 78 


'CHAPTER V. 

PASSING OBSTACLES AND MEASURING INACCESSIBLE 


DISTANCES. 

PASSING OBSTACLES, By Parallel Lines_ 87 

By G0° Angles- 87 

TO MEASURE INACCESSIBLE DISTANCES, By triangles- 88 

Stadia Measures_ 92 

The Gradienter_ 97 


CHAPTER VI. 

PLATTING AND COMPUTING AREAS. 


PLATTING, Instruments used_ 101 

COMPUTING AREAS, Triangles_ 105 

Quadrangles: Rectangles, Trapezoids and Trapeziums— 106 

Irregular Polygons- 107 

Offsets- 108 

Rectangular Coordinates_ 110 

Application to Area_ ill 

The Traverse Table_ 11G 

Meridian Distances_ 118 

Supplying Omissions_ 126 

Reducing Irregular Polygons _ 132 

Division and Partition of Land _;_ 13G 

Method by Approximations_ 14G 

Field Notes. 147 




































TABLE OF CONTENTS. 


VII 


CHAPTER VII. 

CURVELINEAR SURVEYING. 

Preliminary Propositions_ 151 

To run Curves with Picket and Tape_ 152 

Field Notes of Transit Lines_ 154 

To run a Curve with the Transit, Different Methods_ 155 

To locate a Curve from its Middle Point_ 159 

To locate a Curve from some Intermediate Point_ ICO 

To locate a Curve from Point of Intersection_ 1G1 

Passing Obstructions in Line of Curve_ 162 

Compound Curves- 163 

Useful Formula_ igg 


CHAPTER VIII. 

ORIGINAL SURVEYS. 

Surveys, Classified_ 167 

Original Surveys, Government and Private_ 167 

PUBLIC DOMAIN, How and when Acquired_ 168 

Amount of- 170 

Origin of Systems of Surveys of_ 170 

Laws relating to Survey of, where found_ 174 

U. S. LAWS RELATING TO SURVEYS OF PUBLIC LANDS. 

Appointment of Surveyor General- 174 

Qualifications of- 174 

Term of Office_ 175 

When Records and Field Notes to be turned over to 

the State_ 175 

Discontinuance of Office- 175 

When Authority to vest in Com. of Gen. Land Office_ 176 

Free Access to Field Notes and Records_ 176 

Surveyor General to Employ Deputies_ 176 

To cause Survey of Base and Meridian Lines- 177 

To cause Survey of Private Land Claims- 177-190 

To inspect Surveys in Person or by Agent- 178 

Pay of Agent- 178 

Deputy Surveyor to Give Bond- 178 

Deputy Surveyor to make Oath to Field Notes- 179 

Penalty for Fraudulent Survey- 179 

PUBLIC LANDS, How Divided into Townships- 179 

Township Lines, how marked- 180 

•Townships, how subdivided into Sections- ISO 

Sections, how numbered-171-180 

Section Corners, how marked- 180 

Excess or Deficiency over six miles- 180 

Lines, how marked and measured- 181 












































VTTI 


TABLE OF CONTENTS. 


What Surveyors to note in Field Books. 

Disposition of Field Books and making of Plats.... 
SECTIONS AND SUBDIVISIONS OF SECTIONS. 

How Boundaries and Contents are found... 

U. S. Survey Corners the true ones. 

Corners of X and K Sections not set by Government 

Survey.. 

Boundary Lines of U. S. Survey the true ones..... 

Those not run, how found. 

True Contents of Sections returned. 

True Contents of X and K Sections which are not 

Returned. 

Fractional Sections, how divided. 

When ordinary Course may be departed from. 

Surveys in Nevada, Oregon, and California. 

When Rectangular System may be departed from. . 

Instructions, a part of Contract. 

Survey of Mining Claims and Rights of Owners.... 

Appointment of Mineral Surveyors. 

Plats and Field Notes of Mining Surveys. 

Contracts to be approved by Com. Gen. Land Office 

Commissioner to fix Prices for Surveys, etc. 

Extra Price in Oregon, Washington, and California 

Penalty for Interference with Surveys. 

Surveyors appointed to select Timber Lands. 

Duty of Director of Geological Survey. 

INSTRUCTIONS TO SURVEYORS GENERAL. 

Two Mile Blocks, Act of 1796. 

Subdivisions into half Sections, Act of 1800. 

Changes in manner of Subdividing, Double Corners, 

etc. 

How Area of Fractions is Calculated. 

INSTRUCTIONS OF 1894. 

System of rectangular Surveying. 

Establishment of Meridians, Base Lines, and Parallels 

Division into T >wnships and Sections... 

Excess or deficiency in measurement. 

How Townships and Sections are numbered. 

Instruments to be used. 

Tests and Adjustments of. 

Chains and Tally pins... 

Process of chaining. 

Leveling chain and Plumbing pins. 

Marking lines. 

Marking random lines.. 

Insuperable objects in line. 

Witness Points where made. 

Establishing Corners. 

Marking Tools.. 

Descriptions of Corners... 


181 

181 

181 

182 

182 

182 

182 

182 


182 

183 

183 

184 


184 

185 
185 

187 

188 
188 
188 
189 


190 


191 

191 


192 

192 



199 



202 


202 

203 

204 

204 

205 

205 

206 
206 


206 

207 

20 ? 












































TABLE OF CONTENTS. IX 

Abbreviations. 208 

Standard Township Corners, how marked. 208 

Witness Corners, how marked. 211 

Witness Corners in Roads. 211 

Witness points, how marked. 212 

Corners on rock. 212 

Location of Mounds. 212 

Mounds of Stone. 212 

Bearing Trees. 213 

Stones for Corners. 214 

When Lines to be discontinued at Corners. 214 

Orientation of Corners. 214 

Size of Posts, Mounds, etc. 215 

Corner Materials. 215 

Initial Points. 215 

Base Line. 215 

Principal Meridians... 217 

Standard Parallels... 217 

Guide Meridians. 217 

Township Exteriors. 218 

Township exteriors where impassable objects occur. 219 

Method of Subdividing. 219 

Method of Subdividing, Exceptions. 223 

Meandering Streams. 224 

Meandexing Lakes. 225 

Objects to be noted. 226 

Presci’ibed Limits for Closings and Lengths of Lines 228 

Field Notes Blank Books furnished. 229 

What Original Field Notes are. 229 

Description of Old Corners. 229 

Description of Surveys of Base Standard and Merid¬ 
ian Lines. 229 

Description of Exterior Boundaries of Townships.. . 230 

Description of Subdivisions of Townships. 230 

Diagrams. 230 


CHAPTER IX. 

SUBDIVISION OF SECTIONS. 

SUBDIVISION OF SECTIONS. 231 

Four Different Cases. 233 

Quarter Sections. 234 

Half-Quarter Sections. 234 

Fractional Sections. 234 









































X 


TABLE OF CONTENTS. 


Section Six- 235 

Sections made Fractional by Waters.—- 336 

Irregular Subdivision of Sections made Fractional by Waters— 237 

Exceptional Cases- 238 

Private Surveys_ 238 

Highway Surveys_ 239 

Surveys for Town Plats- 239 

What Plats in Michigan must contain- 240 

What Record of Plats in Michigan must contain_ 240 

Monuments_1_ 242 


CHAPTER X. 

RESURVEYS. 

RESURVEYS. 

Authority of Surveyor- 244 

What the Surveyor is called on to do- 244 

DECISIONS OF SUPREME COURTS, Giving- 

Rules for construing Descriptions of Land_ 244 

Adverse Possession- 254 

Rules of Construction when Land borders on Waters— 255 

How to Locate Corners and Boundary Lines_ 262 

General Rules- 263 

Rules Applicable to U. S. Survey_ 280 

Mineral Surveys_ 289 

How to Write Descriptions for Deeds_ 290 


CHAPTER XI. 

RE-LOCATING LOST CORNERS. 

General Rule- 295 

Lost Corners of U. S. Survey in Base Lines, etc._ 296 

Lost Closing Section Corners- 296 

Lost Interior Section Corners- 297 

Lost Township Corners- 297 

Lost Quarter-Section Corners_ 297 

Lost Meander Corners- 297 

Exceptional Methods_ 298 

HOW TO FIND LOST CORNERS, Evidences of 

Original Posts- 299 

Bearing Trees- 300 

Fences- 30 i 

Distant Corners_ 302 

Persons-- 303 




































TABLE OF CONTENTS. 


XI 


CHAPTER XII. 


MISCELLANEOUS. 

Miscellaneous Questions_-_ 304 

RIGHTS, DUTIES, ETC., OF SURVEYORS_ 311 

To fix Lines by Consent of Parties_ 311 

Have no Authority of their own for that purpose_ 312 

Or to determine where Corners and Lines are— 312 

Old Boundaries not to be disturbed_ 313 

County Surveyor’s Certificate not Admissible in evi¬ 
dence in Michigan_ 313 

Surveyor Liable for Damages for Unskillful Work_ 313 

Judicial Functions of Surveyors___-_ 314 


CHAPTER XIII. 

MAP DRAWING AND LETTERING. 


Paper, Pens- 329 

Ink, Instruments- 330 

Execution- 331 

Principles- 332 

Border- 333 

The Meridian- 334 

Scale_ 335 

Lettering- 336 

Characteristics of Letters, Roman Letters- 337 

Proportions of Letters- 338 

Title.-. 341 


CHAPTER XIY. 

LEVELING. 


Definitions_ 343 

Difference between True and Apparent Level- 343 

Instruments for Leveling- 344 

The AVye Level and its Adjustments....— 345 

Leveling Rods, Target and Speaking—-- 349 

To find Difference in Level of Different Points- 351 

Drawing Profile- 356 

Drainage Surveying- 357 

Alphabets--—-*-- 373 






































XII 


TABLE OF CONTENTS. 




TABLES. 


Suggestions to Young Surveyors- i-iv 

Trigonometrical Formulae--- iv-vi 

Table of Logarithms- 1 - 16 

Natural Sines and Cosines- 18- 26 

Natural Tangents- 28- 39 

Logarithmic Sines and Tangents- 40- 84 

Traverse Table- 86 - 91 

Departures_ 92 

Natural Secants- 93- 94 

Azimuth of Polaris at Elongation_ 01 

Gradienter Tables- 95 

Mean Refractions- 96 

Acreage of Open Drains- 97 

Acreage of Tile Drains and Capacity of Tile_ 98 

Azimuths of Tangent- 99 

Offsets from Tangent_ 100 

Minutes in Decimals of a Degree_ 101 

Inches in Decimals of a Foot___ 101 

Radii and Deflections_ 101 

Tangents and Externals of a 1 ° Curve_102-105 

Curve Formulae- 106 


APPENDIX. 

Method of finding the true meridian from observations of 


Polaris at any time when visible. 107 

Government method of abridging field notes. 123 

To find a true meridian or other line by the sun. 124 






























EVE ^ T>T TT.^. L 


OF 

LAND SURVEYING. 


CHAPTEE I. 

1. Definitions. Field Work, &c. 

1. Land Surveying is the art of measuring distances 
and running lines on the earth’s surface to determine 
the boundaries or to ascertain the areas of tracts of land. 
The lines run are not mathematical lines, but are repre¬ 
sentations of them, traced upon the earth’s surface by 
means of various instruments, and marked to the eye 
by chops and notches cut upon trees, or rocks, or by stakes 
or stones set in the ground, or any other means to render 
them visible. 

2. Original Surveys are the surveys which are 
first made for the purpose of locating upon the ground 
the boundaries of tracts of land, and marking them by 
visible objects. This work is called the Field Work. 
A full description of what is done is kept by the sur¬ 
veyor and is called the field notes. The field notes 
furnish the data from which to make a map of the 
land and calculate the area. They also furnish the 
evidence from which to again find and identify the 
boundaries upon the ground. 

3. Resurveys are those which are made for the pur¬ 
pose of finding the boundaries which were marked 
when the original survey was made. 





2 


A MANUAL OF LAND SURVEYING. 


4. The instruments most commonly used in land 
surveying are the Chain and Tape for measuring dis¬ 
tances, and the Picket, Compass , Solar Compass and 
Transit for running lines. 

II. Instruments for Measuring Distances and 

Their Use. 

1. The Chain. The word chain is used to represent 
a distance of 66 feet and also an instrument used for 
measuring distances. The chain in most general use 
for land surveying is that invented by Gunter, and 
known as the Gunter chain. It is 66 feet long and 
divided into 100 equal parts, called links. The chain 
is made of wire, in links somewhat less than eight inches 
long. These are joined by two small, round or oval 
rings at each joint. The length of one of these longer 
links, with the two rings or short links taken together, 
make the distance known as a link. 

The best surveyor’s chains are made of steel 
wire, having the links brazed to prevent stretch¬ 
ing by opening of the joints. Chains have every tenth 
link marked with a brass tag. The tags at the 
end of the tenth link from each end have one point; 
those at the twentieth links have two points; those at 
the thirtieth links have three points; those at the 
fortieth links have four points; while that in the centre 
or fiftieth link is rounded and has no point. Heavy 
chains of iron wire, with open joints, are of little value. I 
It is very difficult to measure correctly with them, over 
rough ground, owing to their weight. They stretch 
rapidly by wear and by the opening of the joints. 
Chains fifty links long are used to measure over rough 
ground. 

2. Chains Stretch by use, chiefly from wear in the 
joints. The best steel brazed chains, when in constant 
use on gritty ground, will stretch six inches or more 
in a year from this cause alone. They may be corrected 
in several ways. They may be shortened a limited amount 







INSTRUMENTS FOR MEASURING DISTANCES. 3 

by turning up the nuts or burrs which hold the handles 
in place. They may be shortened by taking out short 
links or rings. The better way is to distribute the 
correction evenly throughout the chain, by putting 
each link in a vise and striking lightly on the end with 
a hammer, shortening it in that way. 

The links in the chain get bent by use. When many 
of them are bent, the chain becomes elastic and will 
elongate from one to two inches when pulled. Chains 
should be examined before using and the links straight¬ 
ened. They should be frequently compared with a stand¬ 
ard, that their length may be known, and they should 
be kept near the true length. 

3. Steel Tapes are made for the use of land sur¬ 
veyors. They are light, so that they may be readily lev¬ 
eled up in measuring over rough ground or on a slope. 
They do not stretch. There are no links to get kinked 
and thus cause a false measure. They are in every way 
more accurate and convenient than the chain. The best 
tapes for general use are made of the best quality of 
steel ribbon, polished and blued, from % to % of an inch 
wide, and No. 30 to 32 thick. The wider thinner tapes 
are nearly useless for field work. 

Tapes are made of any length and graduated to 
suit the work for which they are designed. A tape 
66 feet long, graduated to links, is best adapted to 
general use. Tapes 50 or 100 feet long, graduated to feet 
and hundredths, are better adapted for use in many cities. 
Tapes from 200 to 400 feet long or even longer are made 
for special uses. With them long lines may be rapidly 
measured with an accuracy fairly comparable with the 
best work of the coast survey. 

Two precautions need to be observed with steel tapes. 
When in use they should be kept out at full length and 
never be doubled on themselves. If doubled they are 
easily kinked and broken. When done up, they should be 
wiped clean and wound on open reels to prevent rusting. 


4 


A MANUAL OF LAND SURVEYING. 


4. A light wire is a cheap and handy substitute for 
the chain or tape. It is necessary to find its length in 
some way and then for even lengths of the wire it is 
capable of as accurate work as the best tape. 

5. Marking Pins are used with the chain and tape 
in measuring. They are usually made of heavy wire 
about 14 inches in length, with one end sharpened to 
stick in the ground and a ring turned on the other end 
for convenience in handling. Strips of cloth are tied 
in the rings so that they can be seen more readily. 
The marking pins used in the United States surveys 
have heavy points, for dropping plumb when chaining 
on slopes. It is convenient to use eleven pins in chain¬ 
ing. One of them is stuck at the starting point, the 
leader takes ten, and then there is always one to start 
from, when the tallies are kept in even tens. 

6. Measuring or chaining. Two men are required 
for this, and a third man can be of great assistance when 
chaining on slopes and accurate work is to be done. 
The care and accuracy required will tlepend on the 
interests at stake. The surveyor would mistake his 
calling who should attempt to measure land worth fifty 
cents an acre with the same care he would use in meas¬ 
uring land worth fifty dollars or more per inch. In 
making measurements the following things are to be 
observed, with greater or less care and accuracy of 
detail, according to the importance of the work in hand. 

1st. Chains are not adapted to great accuracy in meas¬ 
urements. For the best work use a steel tape, of which the 
exact length at a given temperature, and the rate of 
expansion are known. Tapes are usually made to be 
of standard length at a temperature of about 60°, F. 
The rate of expansion by heat varies with the kind and 
quality of steel in the tape. It approximates closely 
to .000007 for each change of a degree in temperature. 
Thus a tape which is 100 feet long at 60° F. will be 
100.014 feet long at 80° F. For very exact measurements 


INSTRUMENTS FOR MEASURING DISTANCES. 5 

take note of the changes in temperature and correct 
for expansion and contraction. A thermometer is 
needed for this. 

2d. Measure in straight lines. In ordinary work, pick¬ 
ets or rods set up along the line, in sufficient numbers 
for the chainmen to range by, will enable them to 
secure as great a degree of accuracy as is required in 
this respect. 

3d. Measure on level lines. To do this the tape may 
be brought to a level line and the successive measures 
transferred to and from the ground by plumb lines. 
Use a plumb having a fine, strong line and a long, well 
balanced, sharp pointed bob. Measure down the slope. 
The rear chainman should hold the tape steadily and 
firmly at the mark, bracing his hand against his leg near 
the ground for a support. The leader brings his end of 
the tape level and in line. If necessary the follower 
directs him in doing this. He then applies the line to 
the point or mark on the tape, with the plumb-bob very 
nearly touching the ground. When he has the proper 
tension on the tape, and the plumb hangs perfectly still 
and true, he depresses the line enough to make a slight 
mark on the ground with the point of the bob, and 
sticks his marking pin beside it. 

Another method of getting the measure on level 
lines is to drive short stakes or hubs along the line at 
every change in the slope of the surface. Small headed 
tacks are driven in the tops of these hubs. The distance 
between the tackheads is then measured along the sur¬ 
face and each measurement recorded. A level is then 
taken showing the difference in hights of these points. 
The length of the level line is found by calculation. 
Between every two hubs we have a right triangle in 
which we have the hypothenuse given by the tape, and 
the altitude given by the level, to find the base. By this 
method the error may be reduced below 1 in 25,000, 




6 


A MANUAL OF LAND SURVEYING. 




4th. The tape must be drawn to the proper tension. 
Tapes are usually tested under a tension of ten pounds 
when supported the entire length. They should be 
further tested to find the amount of additional strain 
required to overcome the sag, when the tape is not 
supported between the ends. This varies, in different 
tapes, from 6 to 12 pounds for a 100 foot tape. The total 
strain in the unsupported tape in measuring should be 
from 16 to 22 pounds. The exact amount is to be found 
for each tape by trial. 


7. The following is the general method of procedure 
in chaining , modified as the circumstances require. We 
will speak of the chainmen as leader and follower. 
The leader takes his end of the chain or tape and ten 
marking pins, and steps briskly in the direction of the 
line to be measured. One pin is stuck at the starting 
point. Just before the leader has the chain drawn 
out at full length, the follower calls “halt,” and places 
his end of the chain in the proper position at the start 
ing point. The leader shakes out any kinks there may I 
be in the chain, straightens and levels it in the line 
brings it to the proper tension and sticks his pin, calling 
“stuck” when he has done so. When the follower hears 
this signal, and not before , he pulls the marking pin 
and both move quickly forward, repeating the opera 
tion until the leader has stuck his last pin or has 
reached the end of the line. When the leader has 
stuck his last pin he calls “tally.” The follower 
drops his end of the chain and brings forward the ten 
pins which he has, and gives them to the leader, who 
counts them to be sure none have been lost 
and then proceeds as before. The follower need not 
return for his end of the chain. The leader will draw 
it forward to him. When the end of the line is reached 
the leader holds his end of the chain at that point 
while the follower drops his end and comes forward 
and ascertains the distance, if any, between the last pin 
that was set and the end of the line. 










INSTRUMENTS FOR MEASURING DISTANCES. 7 

When chaining on slopes which are so steep that the 
whole length of the chain -cannot be leveled at once, the 
leader first draws it forward the whole length and in the 
line. He then drops the chain and all his marking pins 
and returns to a point where he can level a part of the 
chain and measures the distance, sticking one of the fol¬ 
lower’s marking pins to mark the point, the follower then 
drops his end of the chain, comes forward and taking the 
chain at the same point holds it to the mark while the 
leader measures a second section, and so on in succession 
till the end of the chain is reached, where the leader 
sticks one of his own marking pins. It will not often 
be necessary to take any note of the lengths of the parts 
of the chain measured. Observe only to measure to 
and from the same points in the chain, and take care that 
the count is not lost by getting the marking pins im¬ 
properly mixed together. 

The follower should see that his end of the chain is 
correctly and firmly held in its position when measuring. 
He should, when necessary, direct the leader in keeping 
the true line. The leader should see that his chain is drawn 
straight, level, in line, and to a uniform tension. To assist 
him in keeping the line he should observe objects in the 
range, both front and rear. He should see that his 
marking pins are set at the exact point. They should 
either be set plumb or slanting at right angles with the 
line, so that the measure may be taken from the point. 
When a plumb line is used, the latter is the better way. 
Chainmen should s,tep quickly between points, and in 
chaining keep up \yith a man walking at an ordinary 
gait of three miles an hour. The follower must not 
stop the leader by a jerk on the chain. The leader must 
pull steadily when measuring. No jerking on the 
chain should be permitted. 

If there is a difference in the chainmen the best man 
should take the lead. The chaining should always be 
uniform. In many surveys uniformity of measure is 
more important than great exactness. 




8 


4 MANUAL OF LAND SURVEYING. 




Tests made by the author have led him to the conclusion, that, in 
common country surveying with the chain, nothing is gained by level 
ing the chain where the ground slopes less than five in a hundred. He 
finds that in field practice, under the ordinary conditions, more is lost 
by the sag of the chain than is saved by leveling. In one careful field 
test, six links was lost in a mile by leveling the chain, that being the 
net difference in favor of surface measurements for that distance. 

In that class of work, measurements made along the surface may be 
corrected on the ground, as follows: 


When ground slopes 4 in 

100 add 

.1 

link per clu 

G “ 

44 

44 

.2 

44 

44 44 

8 “ 

44 

u 

.3 

44 

44 44 

9 “ 

u 

41 

.4 

44 

44 44 

10 “ 

u 

44 

.5 

44 

44 44 

11 “ 

19. “ 

u 

u 

44 

44 

.G 

. 1 

44 

44 

44 44 

44 44 






8. The student should practice in the field with the 
chain and steel tape until he is entirely familiar with 
their use, and can do accurate and rapid work. He 
should measure between fixed points over sloping or 
uneven ground, and repeat the measures until he can 
secure uniform results. He may be surprised at first 
to find that he does not measure twice alike. It is well 
to drive a small wooden stake at every tally or tenth 
chain, so that in case a marking pin is lost it will not 
be necessary to go back farther than to the first stake 
to remeasure. Beware of errors in counting the links 
less than a full chain. Count from the right end of 
the chain or tape. When the chain is used do not mis¬ 
take the tag, as 60 instead of 40 or vice versa , or count odd 
links the wrong way from the tag. Beware of such mis¬ 
takes as 64 instead of 56, or 48 instead of 52. The tape is 
generally numbered the whole length from 0 to 100. 
Nearly the same care is needed to avoid mistakes in read¬ 
ing as with the chain, especially to read the distance from 
the right end of the tape. Otherwise such mistakes 
as giving the distance 56 instead of 44 are very liable 
to occur. 


III. Measures of Length and Area. 

1. The measures in most general use among surveyors 
are based on the Gunter chain. The surveyor is how¬ 
ever frequently required to express his measurements in 
units of the old linear and square measure. 








MEASURES OF LENGTH AND AREA 9 

Table of Chain Measure. 

7.92 inches or .66 foot=l link. 

66 feetzrlOO links—1 chain. 

80 chains=l mile. 

In country surveying the smaller measures are taken 
in links and parts of a link and distances less than a 
quarter of a link are not counted In the more exact 
work in cities, the foot and its subdivisions are in com¬ 
mon use, and on account of the greater ease in making 
computations upon the decimal system, the plan of 
subdividing the foot decimally is adopted by many 
surveyors, and is growing in favor 

2. Old Linear Measure: 

12 inches = 1 foot. 

3 feet = 1 yard. 

16 % feet = 1 rod. 

40 rods = 1 rood or furlong. 

320 rods — 1 mile. 

Measfres for Area 

3 Chain Measure: 

100,000 square links, or ) . nprp 

10 square chains $ dL 

640 acres = 1 sq mile or section. 

36 sections = 1 township 

In the United 
States land sys¬ 
tem, the square 
mile is known as 
the Section. It is 
subdivided into al¬ 
iquot parts, which 
are described ac¬ 
cording to their 
place in the sec¬ 
tion. The manner 
of naming these 
subdivisions of a 
section is indicat¬ 
ed in Figure 1. 


WE 5 T 


j. 


N.E.J 




-N 

a F 

Uj 

in 


D 

S.E--k 

dru 

of 

£ 



Fig. i. 







10 


A MANUAL OF LAND SURVEYING. 


When, because of lakes, rivers, reservations, adjacence 
to township boundaries, or other causes, any of the parts 
of a section are increased or diminished from their 
normal amount, they are known and described as Frac¬ 
tional. That word is used to indicate that the tract to 
which it is applied is not one of the regular subdivisions 
of the section. When a fractional lot is small it is the 
custom of the United States land department to attach 
it to, and sell it with, an adjacent larger tract which gives 
the name to the description of the whole tract. The 
manner of describing fractional lots is indicated in Fig¬ 
ure 2. It is also a custom to number the fractional lots 


on the plats and 
describe them by 
numbers, as for 
example, Lot No. 
3 of Section 18. 
The latter method 
requires a refer¬ 
ence to the plat to 
know the location 
of the lot, while 
the former method 
does not. 



*4 $3 



I Old English Land Measure: 

144 square inches = 1 square foot. 

272f£ square feet = 1 square rod. 

40 square rods = 1 rood. 

100 square rods = 1 acre. 

Square rods and feet are still in common use as sub¬ 
divisions of the acre. The rood and furlong are very 
nearly if not quite obsolete in the United States. 

5. Spanish Measures. — In Spanish colonies in 
America, the Spanish system of land measures was used 










MEASURES OF LENGTH AND AREA 


11 


in describing and measuring the land grants, and has 
continued in use down to the present time in a large 
extent of country. The principal unit of measure is the 
“vara,” which seems to be a somewhat variable one. In 
a report of the 14tli of November, 1851, from the surveyor- 
general of California, it is stated that all the grants, etc., 
of lots or lands in California, made either by the Spanish 
government or that of Mexico, refer to the “vara” of 
Mexico as the measure of length; that by common con¬ 
sent, in California, that measure is considered as exactly 
equivalent to thirty-three American inches. That officer 
enclosed a copy of a document he had obtained as being 
an extract of a treaty made by the Mexican government, 
from which it would seem that another length is given to 
the “vara;” and by el. H. Alexander’s (of Baltimore) Dic¬ 
tionary of Weights and Measures, the Mexican vara is 
stated to be equal to .92741 of the American yard. The 
general land office, however, has sanctioned the recog¬ 
nition, in California, of the Mexican vara as being 
equivalent to 33 American inches. 

Extract of a treaty made with the Mexican government, which accom¬ 
panied a report dated November 14, 1851 , from the U. S. surveyor- 
general of California, respecting the ratio of land measures between 
those employed under the Mexican government and those in use in 
the United States 


[From the Mexican ordinance for land and sea.] 


Article 20th of the agreement entered into between the minister pleni¬ 
potentiary of the Mexican government and her agents in London, 
the 15th of September, 1S37, with the holders of Mexican bonds. 

20th. In compliance of what is ordered by the seventh article of 
the preceding law, and in order to carry into effect the stipulation in 
the preceding agreement in regard to the holders of bonds deferred, 
it is declared that the act of which mention is made in said agreement 
answers to 4840 English yards squared, equivalent to 5762.403 Mexican 
varas square; inasmuch that the “sitio de ganado moyer” contains 
4338.464 acres, the Mexican vara having been found by exact measures 
equal to 837 French millimetres. 

Deducing the ratio of 4840 square yards 
and 5762.403 square varas, the vara 


will be. 

Deducing the 4338.464 acres 







12 A MANUAL OF LAND SURVEYING. 


Names of the 
Measures. 

Figures of 
Measures. 

Length of figures 

expressed in 

varas. 

Breadth in varas. 

1 

Areas in square 

varas. 

Areas in cabal- 

lerias. 

Sitio de ganado moyer 

Square- 

5,000 

5,000 

25,000,000 

41.023 

Criadero de ganado 






vnnypr 

do 

2,500 

2,500 

6,250,000 

10.255 

Sitio Vie ganado menor 

do. _ 

3,333% 

3'333% 

11,111,111 

18.232 

Criadero de ganado 






ni prior 

do 

1,666§ 

1,666% 

2,777,7772 

4.558 

Caballeria de tierra— 

Right angled 




parall’gram 

1,104 

552 

609,408 

1 

Media, caballeria 

Square 

552 

552 

304,704 

y 2 

Cuarto caballeria o 





Suerte de tierra- 

Right angled 






parairgram 

552 

276 

152,352 


Fenega de sembra- 






dnro de maiz 

do 

276 

184 

50,784 

1-12 

Sa l a. pa ra, casa 

Square 

50 

50 

2,500 

0.004 

Fundo legal para pue- 





bios 

do. 

1,200 

1,200 

1,440,000 

2.362 


The Mexican vara is the unit of all the measures of 
length, the pattern and size of which are taken from the 
Castilian vara of the mark of Burgos, and is the legal 
vara used in the Mexican republic. Fifty Mexican varas 
make a measure which is called “ cordel,” which instru¬ 
ment is used in measuring lands. 

The legal league contains 100 cordels, or 5,000 varas, 
which is found by multiplying by 100 the 50 varas con¬ 
tained in a cordel. The league is divided into two halves 
and four quarters, this being the only division made of it. 
Half a league contains 2,500 varas, and a quarter of a 
league 1,250 varas. Anciently, the Mexican league was 
divided into three miles, the mile into a thousand paces 
of Solomon, and one of these paces into five-thirds of a 
Mexican vara; consequently, the league had 3,000 paces of 
Solomon. This division is recognized in legal affairs but 
has been a very long time in disuse—the same as the pace 
of Solomon, which in those days was called vara, and was 
used for measuring lands. The “mark” was equivalent 
to two varas and seven-eighths—that is, eight marks com 
































MEASURES OF LENGTH AND AREA. 13 

taining twenty-three varas—and was used for measuring 
lands 

In Texas the surveys are made on the vara system. A 
20-vara chain is used, the area calculated in varas, and 
when necessary reduced to acres. The held notes contain 
no system of measurement except varas. Nearly all the 
old leagues were laid off in rectangular form, and nearly 
all the subdivisions since have been by lines parallel with 
the original league lines. 

The following table of comparisons gives the system of 
land measures in use in that state: 

1 vara = 33% inches. 

1900.8 varas == 1 mile. 

25,000,000 sq. varas = 1 league = 4428.4 acres. 

1,000,000 “ “ == 1 labor = 177.136 “ 

5645.376 “ “ = 1 “ 

1 “ “ = .000177 “ 

6. Old French Measures were used in laying off 
land in the French colonies, and still find a place in some 
parts of the country. The unit was the “arpent,” of 
which there were different values, varying from three- 
fourths of an acre to an acre and a half. The “ arpent 
d’ordonnance” or legal arpent equalled 1.262 acres, and 
contained 100 square perches of 22 “ pieds du roi ” on a 
side. 

The old French linear measures were the old Paris foot 
called “ pied du roi ” and its sub-multiples— 

12 points = 1 ligne. 

12 ligne = 1 ponce. 

12 ponce = 1 pied du roi = 12.789 inches. 

6 pieds du roi = 1 toise,—interesting as being the unit 
employed in the survey of the great French meridian arc, 
on which the metre was founded. 

Modern French measures are upon the Metric System. 



14 


A MANUAL OF LAND SURVEYING. 


7. Standard Measures. 

The constitution of the United States says that con¬ 
gress shall have power to establish a system of weights 
and measures. It has, however, never done so. In 1832 
the secretary of the treasury assumed the authority to 
adjust and regulate the weights and measures in use in 
the custom houses, and delegated the construction and 
adjustment of standards to Mr. Hassler, who was then 
superintendent of the coast survey. 

The standard of length adopted was a yard, as meas¬ 
ured between the 27th and 63rd inches of a scale made in 
London, by Troughton, and brought to this country in 
1814. This scale is a copy of the old British Standard, 
known as the Bird Standard of 1760. 

At a temperature of 59.62° F. it is equal in length to 
the Imperial Standard at 62° F. Although Congress 
never adopted that yard as a standard, it authorized the 
transmission of copies thereof to the several states. In 
many of the states these copies have been legally adopted 
as the standards. Other states have no legal standards. 
The Michigan standard is a brass yard, of exact length 
at a temperature of 58.40° F. It is both a line and an end 
measure. It is doubtful if these standards in the several 
states are kept in such a manner as to be reliable for 
purposes of comparison or if they are so kept, whether 
the officers in charge of them have the skill and the 
facilities required for making accurate comparisons. 
Standard rods are sold by dealers but they are more or 
less discrepant in length. Surveyors who desire to know 
the true length of their standard measures can send them 
to the Superintendent of the Coast and Geodetic Survey, 
at Washington, who will cause them to be compared and 
the government stamp placed on them, giving their 
exact length. The examination and test, for which a 
fee of fifty cents is charged, secures a sufficient degree of 
accuracy for ordinary purposes of the surveyor. Where 
an extra degree of accuracy is called for a higher fee is 
charged. 




MEASURES OF LENGTH AND AREA. 15 

Although Congress has not adopted a general standard 
of measure, it has adopted a standard for the measure¬ 
ment of the public lands, which so far as the resurvey or 
subdivision of those lands is concerned is tinal. In sec¬ 
tion 2395 of the revised statutes of the United States, it 
is enacted that “all lines shall be measured with chains 
containing two perches of sixteen and one-half feet, each 
subdivided into twenty live equal links. In section 2396 
it is enacted that “All the corners marked in the surveys 
returned by the Surveyor General shall be established as 
the proper corners” &c.; and that “the boundary lines 
actually run and marked in the surveys returned by the 
Surveyor General, shall be established as the proper 
boundary lines of the sections and subdivisions for which 
they were intended, and the length of such lines as 're¬ 
turned shall he held and considered as the true length 
thereof.” 

This enactment makes an actual standard of measure 
between every two adjacent corners of the government 
survey, which is the only legal standard for measures of 
that line. The surveyor, in resurveying or subdividing 
the public lands, has thus a standard laid dow r n for him 
on every line previously run by the government deputy 
surveyor and has only to adjust his chain to that stand¬ 
ard. This is practically done on the ground by apportion¬ 
ing any difference between the surveyor’s measure of a 
given line and the length of the line as returned in the 
field notes pro rata between its different parts. 

Example— It is required to locate the half-quarter cor¬ 
ner on the line described in the field notes as running, 
“West on corrected line between Sections 11 and 14 
39.72, set qr. sec. post,” etc. 

Suppose the surveyor on measuring this line finds the 
distance between the two corners, as actually marked on 
the ground, to be by his chain 39.84 chains. Then his 
chain is too short and its legal length for that line is to 
its nominal length as 39.72 is to 39.84 and the distance to 
the half-quarter corner is by the new measure 19.92 chains. 



16 


A MANUAL OF LAND SURVEYING. 


IV. Instruments for Running Lines and 

Their Use. 

1. The instruments most commonly used in running 
lines are the picket , the compass and the transit. There 
are various modifications of the compass and transit. 
The methods of running lines with these instruments 
will be treated of in connection with the description 
of them. 

2. The Picket or Rod is the simplest device for 
ranging lines. It is simply a straight rod an inch 
or two in diameter and having a sharp point to stick in the 
ground. The author prefers to have them sharpened to 
a long slim point at the top also, and that the pickets shall 
be of such a length as to be the height of the eye when 
firmly planted in the ground. Where timber is plenty 
they may be cut from small straight saplings, or split 
from body wood as they are wanted, and left standing 
where they are used, as a guide to the chain men. 

3. To range a line with pickets. Set the first picket 
at the starting point and a second a short distance 
away in the direction in which the line is to run. 
Then go ahead and set picket after picket at such 
distances apart that at least three of them can be 
distinctly seen at the same time. Set the pickets plumb 
and align them by sighting over the sharpened points 
at the top. A plumb line will be of assistance in rang¬ 
ing lines over uneven ground. Set short stakes in the 
line at uniform distances apart. Then if the line was 
intended to strike a particular point and missed, it 
may be corrected by measuring the perpendicular dis¬ 
tance from the line to the point, and then moving each 
intermediate stake its proportional part of that dis¬ 
tance according to the distance it is from the starting 
point. 

Example 1.—Commencing at the southwest corner of 
Mr. B.’s farm, I ran north, setting stakes on the trial 
line every ten chains. At 40.00 chains, my line inter- 




INSTRUMENTS AND THEIR USE. 17 

sected the north line of his farm 32 links east of his 
northwest corner. What correction must be made for 
each stake ? 

Solution .—The first stake being set at 34 the distance 
between points must be corrected 34 of 32 = 8 links, and 
as the trial line came out to the east of the corner, the 
stakes on that line must be moved to the west. The 2d 
stake being at 34 the distance between points must be 
moved west % of 32 = 16 links. Similarly the 3d stake 
must be moved west 24 links. 

NOTE.—Sections of the United States survey are tracts of one mile 
square. Monuments are set at each corner called Section Corners. 
Others are placed midway between them on the section lines called 
quarter posts or quarter section corners. Some sections greater or 
less than these are called Fractional Sections. 

Example 2 —Commencing at a point 12 links west 
of the quarter post in the south side of Section 20, I 
ran north, setting stakes on the trial line every ten 
chains. At 80 chains my line intersected the north line of 
the section, 36 links west of the quarter post. What cor¬ 
rection must be made to place the intermediate stakes 
in the true line between the quarter posts, known as 
the quarter line ? 

Answer .—Commencing with the first ten chain stake 
they must be set east, 15, 18, 21, 24, 27, 30, and 33 links 
respectively. 

Example 3.—Commencing at a point 24 links west 
of the southwest corner of section 16, I ran a trial 
line north, setting stakes every ten chains. At 80.36 
chains, the line intersected the north line of the section, 32 
links east of the section corner. What is the correction 
to be made at each stake to place it in the true section 
line and at the equidistant points? Answer to be found 
by the student. 

NOTE.—This solution requires corrections both for line and meas¬ 
ure. It is a cardinal principle of land law that the original measure¬ 
ments and monuments which were made in tbe survey in accordance 
with which the land was sold are in law the true measures and monu¬ 
ments. All subsequent measures for tin* purpose of locating bound¬ 
aries must be made to^conform with the original measures. 


3 


18 A MANUAL OF LAND SURVEYING. 

Trial or random lines, as they are usually called, are 
often run one side of the true line, purposely to avoid 
obstacles, like fences and hedge rows. The surveyor, by 
a judicious selection of ground for the random line can 
often save a great deal of labor and time of the party, 
by avoiding obstacles which would otherwise have to be 
removed or offset around. Randoms from which the 
true line is to be found should be run with as great care 
as any line. 

The student should practice running and measuring 
trial lines between points until familiar with the pro¬ 
cesses. He should run various randoms to find the line 
between the same points and see how they agree when 
corrected for true line. 

4. To range a true line between points that can not be 
seen from each other but can both be seen from some inter¬ 
mediate point, as a lull . 

Set up Hags at the two points. Two persons then 
take pickets and station themselves, a short distancer 
apart, at the intermediate position from which the flags 
can be seen. They face each other and each in turn 
aligns the other between himself and the flag toward 
which he faces, until the true line is reached, when the 
pickets are set in the line. 

5. To pass obstacles in the line. 

From the last two pickets preceding the obstacles, set 
two other pickets on a line parallel with [the true line 
and at a sufficient distance to pass the obstacle. Prolong 
the parallel line far enough to set two pickets beyond 
the obstacle and then regain the original line by meas¬ 
uring back from these two pickets. 

6 . The methods of running lines witli the compass 
and transit will be given in connection with the descrip¬ 
tions of these instruments. 







DESCRIPTION OP INSTRUMENTS. 


19 


CHAPTEK II. 


Description of Instruments. 

1. The Surveyor’s Compass. The essential fea¬ 
tures of the surveyor’s compass are a magnetic needle for 
finding a meridian line, a circle graduated to half degrees 
known as the limb, for laying off angles from the 
meridian, and sights attached for use in prolonging lines 
on the ground. 

When the limb and sights are on separate plates move- 
able upon each other around a common center through 
an arc of 15° or 20°, and a vernier is attached, the instru¬ 
ment is known as the Vernier Compass. 

The use of the vernier is chiefly for setting the sights 
of the instrument so that they will be in the true north 
and south line when the magnetic needle points to zero 
on the limb. There is only a small portion of the earth’s 
surface in which the needle points to the true north. 
A line passing through those places where the needle 
points truly north is called the agonic line or line of no 
variation. This line runs in a northerly course and is 
constantly changing its position. At all places outside 
the line of no variation, the needle points to the east 
or west of true north. This difference between the 
direction of the needle and the true meridian is spoken 
of as the variation , or, more correctly, the declination 
of the needle. The vernier is used to measure the angle 
between these two lines. 


20 


A MANUAL OF LAND SURVEYING. 



Fig. 3.— VERNIER COMPASS-G-Inch Needle. 

Sometimes there is added a divided circle or limb with 
verniers by which angles can be taken throughout the 
entire circle independently of the needle. The instrument 
in this form is called the railroad compass. The addition 
of leveling screws and a revolving telescope in place of 
the plain sights makes a surveyor’s transit of it. 






















ADJUSTMENTS OF THE COMPASS. 


21 


The Plain Compass consists of a circular box of 
brass, usually about six inches in diameter, resting upon 
an arm of the same metal about fourteen inches in length* 
At the extremities of the arm are vertical attachments 
through which are fine slits, terminated at intervals by 
circular apertures, which serve as sights in directing the 
instrument upon any point. At the centre of the box is 
a small vertical pin upon which is balanced a slender 
magnetized bar of steel, called the Needle. 

Turning with a free horizontal motion, the pointed 
ends of the needle traverse the graduated circumference 
of the circle. The plane of the sights passes through the 
center of the circle and cuts the circumference in two 
points marked N and S, otherwise distinguished as the 
north and the south points of the instrument. From 
these points the graduation of the circle runs 90° in each 
direction to the points marked E and W. 

A circle of plate-glass forms the cover of the box. 
Two small spirit levels are placed at right angles to each 
other upon the arm, to aid in rendering the plane of the 
instrument horizontal. 

The compass is mounted upon a three-legged support 
called a Tripod, or upon a single staff called a Jacob 
Staff, with which it is so connected as to admit of being 
turned in any desired direction. In using the compass, 
the surveyor should keep the south end toward his per¬ 
son, and read the bearings from the north end of the 
needle. lie will observe that the letters E and W on the 
face of the compass are reversed from their natural 
position, to correspond with the line of the sights, in 
order that the direction may be correctly read. 

II. Adjustments of the Compass. 

The Sights of the compass should be truly at right 
angles with the plate, so that when set up and leveled 
ready for use the line of sight will be in a vertical 
plane. 



22 


A MANUAL OF LAND SURVEYING. 




The needle should cut opposite degrees in any part of 
the circle, and should have its ends in line with the 
centre. 


The levels should be parallel to the plane of the plate. 
To adjust the compass to these conditions begin with 


The Levels.— First bring the bubbles into the centre, 
by the pressure of the hand on different parts of the plate, 
and then turn the compass half-way around; should the 
bubbles run to the edge of the tubes, it would indicate 
that those ends were the highest; lower them by tight¬ 
ening the screws immediately under, and loosening those 
under the lowest ends until, by estimation, the error is 
half removed; level the plate again, and repeat the first 
operation until the bubbles will remain in the centre, 
during an entire revolution of the compass. 


The Sights may next be tested by observing through 
the slits a fine hair or thread, made exactly vertical by a 
plumb. Should the hair appear on one side of the slit, 
the sight must be adjusted by filing off its under surface 
on that side which seems the highest. 


The Needle is adjusted in the following manner: 
Having the eye nearly in the same plane with the grad¬ 
uated rim of the compass-circle, with a small splinter of 
wood or a slender iron wire, bring one end of the needle 
in line with any prominent division of the circle, as the 
zero, or ninety degree mark, and notice if the other end 
corresponds with the degree on the opposite side ; if it 
does, the needle is said to “cut” opposite degrees ; if not, 
bend the centre-pin by applying the small brass wrench, 
furnished with the compass, about one-eighth of an inch 
below the point of the pin, until the ends of the needle 
are brought into line with the opposite degrees. 

Then, holding the needle in the same position, turn the 
compass half-w r ay around, and note whether it now cuts 
opposite degrees; if not, correct half the error by bend¬ 
ing the needle, and the remainder by bending the centre 


pm. 






adjustments of the compass, 23 

The operation should be repeated until perfect rever¬ 
sion is secured in the first position. 

This being obtained, it may be tried on another quarter 
of the circle ; if any error is there manifested, the correc¬ 
tion must be made in the centre-pin only, the needle 
being already straightened by the previous operation. 

When again made to cut, it should be tried on the other 
quarters of the circle, and corrections made in the same 
manner until the error is entirely removed, and the needle 
will reverse in every point of the divided surface. If the 
needle has lost its polarity, and needs to be remagnetized, 
this is effected in the following manner : 

The operator being provided with an ordinary perma¬ 
nent magnet, and holding it before him, should pass with 
a gentle pressure each end of the needle from centre to 
extremity over the magnetic pole, describing before each 
pass a circle of about six inches radius, to which the 
surface of the pole is tangent, drawing the needle towards 
him And taking care that the north and south ends are 
applied to the opposite poles of the magnet. 

Should the needle be returned in a path near the mag¬ 
netic pole, the current induced by the contact of the 
needle and magnet, in the pass just described, would be 
reversed, and thus the magnetic virtue almost entirely 
neutralized at each operation. 

When the needle has been passed about twenty-five 
times in succession, in the manner just described, it may 
be considered as fully charged. 

A fine brass wire is wound in two or three coils on the 
south end of the needle, and may be moved back or 
forth in order to counterpoise the varying weight of the 
north end* 

The Centre-Pin. — This should occasionally be ex¬ 
amined, and if much dulled, taken out with the brass 
wrench, already spoken of, or with a pair of pliers, and 
sharpened on a hard oil-stone—the operator placing it in 
the end of a small stem of wood, or a pin-vise, and deli- 


24 


A MANUAL OF LAND SURVEYING. 


cately twirling it with the fingers as he moves it back 
and forth at an angle of about 30 degrees to the surface 
of the stone. 

1 

When the point is thus made so fine and sharp as to be 
invisible to the eye, it should be smoothed by rubbing it 
on the surface of a soft clean piece of leather. 

Electricity.— A little caution is necessary in handling 
the compass that the glass covering be not excited by the 
friction of cloth, silk, or the hand, so as to attract the 
needle to its under surface. 

When, however, the glass becomes electric, the fluid 
may be removed by breathing upon it, or touching differ¬ 
ent parts of its surface with the moistened finger. 

III. To Run a Line with the Compass. 

Set up the instrument at the point from which the line 
is to run ; level the plate ; turn the sights in the direction 
in which the line is to run, which may be ascertained by 
the needle or otherwise, as is most convenient. An assist¬ 
ant, known as the rodman or flagman, goes ahead with a 
sharp pointed rod or flag pole to such a distance as is 
convenient, and, guided by the signals of the compass- 
man, sets his rod in line. When the ground is uneven, the 
rodman should select his point at the summit of rising 
ground, when possible to do so, in order to save unneces¬ 
sary setting of the compass. lie should always select the 
point most favorable for setting up the instrument, both 
to get a clear spot for the instrument and to get the best 
point for taking the next sight. 

When setting his rod he should face the compass, hold¬ 
ing the rod plumb and directly in front of him. He 
should move steadily in the direction indicated by the 
signals and not stick the rod down until he receives the 
signal to do so. After sticking it he should look for 
further signals, lest a change in its position might be 
required. After the rod is set the compassman should 
examine his instrument to see that it is in position, cor- 






TO PASS OBSTACLES IN THE LINE. 


25 


recting it and resetting the rod when necessary. He then 
sets up a picket in line near his instrument, to be used for 
aback sight, and moves his compass forward in the line 
to the point marked by the rodman, sets it up in the line, 
with the sights ranging back to the backsight, and con¬ 
tinues the line as far as desirable. The needle may or 
may not be used, according to circumstances. At the 
beginning of the line the direction will usually be obtain¬ 
ed from the needle. If used afterwards on the same line, 
care should be taken to have it in proper condition and 
working freely. When being carried the needle should 
be raised off the pivot, otherwise the point of the pivot 
will become dulled and the needle will not traverse freely. 

IV. To Pass Obstacles in the Line. 

1. When the obstacle is a tree, and no great degree of 
accuracy is required, make a mark on the tree where the 
line strikes it and set the compass up on the opposite 
side of the tree, putting it in line by taking a backsight 
on the tree, and finding the direction of the line by the 
needle. 

2. Make an offset far enough to pass the obstacle 
on a parallel line, the same as when running a picket 
line. When it is found that the line strikes a tree too 
large to be removed, set the rod in line near the tree, and 
then before moving the compass, set the picket for back¬ 
sight at one side of it, a sufficient distance to pass the 
tree. Then move the compass ahead and set it up the 
same distance, and direction from the rod that the back¬ 
sight picket w T as set from the compass. Get the direction 
of the line by ranging to the backsight. Prolong the 
parallel line beyond the obstacle and regain the true line 
in a similar manner. Other methods of passing obstacles 
in line will be given further on. 

Y. The Magnetic Needle. 

1. The compass, because of its being so convenient for 
use has been for many years the principal instrument used 


26 A MANUAL OF LAND SURVEYING. 

in Land Surveying. It is now very generally superseded 
by other instruments in surveys where accuracy is re¬ 
quired. So far as the direction of lines is concerned, all 
compass surveying is based on the tendency of the 
magnetic needle to adjust itself to the magnetic meridian 
when free to do so, in other words to point north and 
south. It is however constantly changing its direction. 

2. Secular Change. The line of no variation, as it is 
commonly called, otherwise known as the agonic line 
seems to have a periodical motion, back and forth, to the 
east and west, like the swinging of the pendulum. The 
length of the period is unknown but probably covers sev¬ 
eral centuries. 

In the United States, so far back as known, its motion 
was to the eastward until the beginning of the present 
century, since which time it has been moving westward. 
In Michigan the secular change has been between 3' 
and U per year to the westward for the past sixty 
years. The agonic line is, in 1890, in the vicinity of 
Lansing. 

3. Diurnal Change. The needle when undisturbed 
and free to move, swings back and forth each day 
through an arc varying from 5 / to 20' or more in amount. 
In the northern hemisphere the north end of the needle 
moves westward from about 8 a. m. until about 1:30 p. m., 
then returning and reaching its former position at about 
8 r. m. The amount of this motion is not uniform from 
day to day, being least on cloudy days ; nor from month 
to month, being least in winter. Nor is it the same in 
different localities. The effect of the diurnal variation is 
such that if a surveyor w T ere to start a line in the morning 
and continue running it all day in the same direction, as 
shown by the needle, he would run a line like a letter S. 

4. Irregular Changes. The needle is subject to sudden 
and violent changes in its direction, sometimes coinci¬ 
dent with a thunderstorm or an Aurora Borealis,—often 
without any apparent cause. The writer has observed a 


THE MAGNETIC NEEDLE. 


27 


change of half a degree in less than ten seconds of time, 
for which there was no apparent or discoverable cause. 
It was supposed to have been occasioned by a magnetic 
storm. 

5. Local Attraction. Iron ore in the earth, or iron or 
steel in the vicinity of the needle will deflect it from its 
normal direction. High mountains or running streams 
are also said to deflect the needle more or less. Pocket 
knives and steel watch chains are prolific sources of error 
as well as chains and axes. 

6 . Difference in Instruments. It is found by obser¬ 
vation that different instruments do not indicate the 
same declination of the needle when observed at the 
same time and place. A difference of 15' is not uncom¬ 
mon. Mr. Hurley made six needles taking great pains to 
have them as nearly alike as possible. He tried them in 
succession on the same centre-pin. Three of them gave 
the same results. The other three differed from 5' to 1CK. 

7. Things to he Observed in Running Compass Lines. 
For these reasons it is practically impossible to run a true 
line and repeat it, relying on the needle alone for direc¬ 
tion. Hence in all original surveys, made with the com¬ 
pass, the field notes of the survey should give the date, and 
state whether the directions of the lines are given accord¬ 
ing to the magnetic meridian. If not, state what the 
angle is between the magnetic meridian and the meridian 
adopted for the survey, or in other words state the decli¬ 
nation of the needle, estimated or allowed for in the sur¬ 
vey. The meridian adopted will usually be as nearly 
coincident with the true meridian as known. Back¬ 
sights should be used whenever the line is prolonged 
beyond a single sight, both to secure accuracy in the line, 
and as a check against local disturbances of the needle. 
They also save time, as a compass can be pointed to a 
backsight in much less time than it takes a good needle 
to settle. 



28 A MANUAL OF LAND SURVEYING. 

8 . Marking Lines. It is a cardinal principle of com¬ 
mon law, as well as the statute law of the United States 
with reference to the public lands, that the original 
surveys as marked on the ground, in accordance with 
which the land was sold, are conclusive as to the corners 
and boundary lines. When the land is once sold, no 
change can be made in the marked boundaries without 
disturbing the vested rights of the owners. Resurveys are 
made to find the location on the ground of the original 
survey. The compass is a useful assistant in pointing 
out where to look for the more certain evidences, such as 
marked trees, stakes or comer stones, and, in the absence 
of anything better, may be used to determine the location 
of the line. A marked tree of the original survey is, 
however, better evidence of the location of the line than 
any line afterward run by a compass. It is possible that 
the line might be exactly retraced by the compass, but 
it could not be known to be so without the aid of other 
evidence. Hence the marks on the ground which define 
boundary lines cannot be made and kept too plain and 
permanent. The field notes and records which describe 
these marks should be full, clear and concise. 

VI. True Meridians and how to Find them with 

the Compass. 

In a country that has had the first surveys made 
and boundary lines marked, and subsequent surveys are 
based on these lines, it is very rarely of any consequence 
to the surveyor to know where the true meridian is. The 
original boundary lines are unchangeable, and it is no 
help to the surveyor to know where the true meridian is 
unless he also knows that the original surveys were in 
conformity with it, and that the causes of error hereto¬ 
fore mentioned can be eliminated. That is very rarely the 
case. His main concern is to know where the lines were 
and not where they ought to have been. The writer in 
nearly a quarter century of active practice as a surveyor 
has never had occasion, except as a matter of curiosity, to 
know where the true meridian was. In making the 
first surveys of a country with a compass, it is well to 


TO FIND A TRUE MERIDIAN. 


29 


know the position of the true meridian, in order that the 
lines may be run as nearly in conformity with it as the 
limitations of the instrument will permit, or that the 
divergence may be known. Subsequently, a knowledge 
of the changes in the declination of the needle is all that 
serves any practical purpose. This can be learned by 
observations on any line between two permanent points. 

To find a true north and south line by means of the 
north star. 

The north star appears to describe a small circle about 
the true north point or pole as a center. The radius of 
this circle is called the Polar Distance of the star. 
This polar distance is not a constant quantity, but be¬ 
comes about % of a minute of arc less every year. On 
the first of January, 1890 it was about 1° 16 / 41". 

When in its revolution, the star is farthest from the 
meridian, it is said to be at its greatest eastern or 
western elongation. 

The times of the elongations as given by a correct 
clock, for latitude from 38° N to 00° N and for the year 
1890, are approximately as shown in the following tables: 


EASTERN ELONGATIONS. 


Day. 

Apr. 

1 

May. 

June. 

July. 

Aug. 

Sept. 


H. 

M. 

H. 

M. 

H. 

M. 

B. M. 

H. 

M. 

H. 

M. 

l 

6 37 

A.M. 

4 39 

A.M. 

2 37 

A.M. 

12 39 A.M. 

10 37 

P.M. 

8 36 

P.M. 

7 

6 14 

44 

4 16 

44 

2 14 

44 

12 16 “ 

10 14 

44 

8 12 

44 

13 

5 50 

44 

3 52 

44 

I 50 

44 

li 52 P.M. 

9 50 

44 

7 48 

4 

19 

5 26 

44 

3 28 

44 

1 26 

44 

K29 “ 

9 27 

44 

7 25 

44 

25 

5 03 

44 

3 05 

44 

1 03 

44 

11 05 “ 

9 03 

44 

7 01 

44 



WES' 

FERN ELONGATIONS 

• 




Day. 

Oct. 

Nov. 

Di 

C. 

Jail. 

Feb. 

Mar. 


H. 

M. 

B. 

M. 

B. 

M. 

B. M. 

H. 

M. 

H. 

M. 

1 

6 2? 

A.M. 

4*25 

A.M. 

2 28 

A.M. 

12 26 A.M. 

10 24 

P.M. 

8 30 

P.M. 

7 

6 04 

44 

4 02 

44 

2 04 

44 

12 02 “ 

10 00 

44 

8 06 

44 

13 

5 40 

44 

3 38 

44 

1 40 

44 

11 39 P.M. 

9 36 

44 

7 43 

44 

19 

5 17 

44 

315 

44 

1 17 

44 

It 15 “ 

9 13 

44 

7 19 

44 

25 

453 

44 

2 51 

44 

12 53 

44 

10 51 “ 

8 49 

44 

655 

44 











































30 A MANUAL OF LAND SURVEYING. 

To find the meridian of a place by means of an elonga¬ 
tion of the north star requires the arrangement of the 
following preliminaries. 

Set two posts firmly in the ground about three feet 
apart east and west, and saw them oft to a level about 
three feet from the ground. 

Lay upon the posts a plank 3 or 4 feet long and 6 or 8 
inches wide, planed smooth on the upper surface, and 
nail or pin it securely to the supports, forming a sort of 
table. 

To the north of the table at a distance of 10 or 12 feet 
set in the ground a stiff pole 12 or 15 feet high, having a 
cross bar nailed to its top, in an east and west direction, 
from which to suspend a plumb-line nearly reaching the 
ground, and having a bob weighing 1 or 2 pounds, which 
may be caused to hang in a pail of water, to insure stead¬ 
iness. 

Provide also a block or piece of plank 8 or 10 inches 
long, and smooth on the under side. Let one of the com¬ 
pass sights be fastened at right angles with the upper 
surface of the block and even with the side which is to be 
toward the south. 

Everything being in readiness, the observer, a few 
minutes before the time of an elongation as given in 
the above Table, should be at his post and begin moving 
the block, even with the south edge of the table, keeping 
the plumb-line and star, as seen through the vertical slit, 
constantly in range with each other. A light will 
generally be needed near the plumb-line, to render it 
visible. As the star approaches its elongation, it will 
appear to move nearly vertical for several minutes, so as 
to be seen without moving the sight. When it is certain 
that the star has reached its elongation, confine the block 
carefully, by sticking a few tacks along its edges. Pro¬ 
ject the vertical slit to the ground by means of a plumb- 
line and mark the point by setting a substantial stake 
with its top a little below the surface of the ground. 


TO FIND A TRUE MERIDIAN. 


31 


Being still careful not to move the block, let an assist¬ 
ant take one of the iron-pointed rods, or a stake, with a 
light, and go a hundred feet or more toward the star, and 
having found the point as directed by the observer, in 
range with the plumb-line as seen through the slit, let 
him mark it by driving a stake. 


Having now two stakes in range of the elongation, the 
remainder of the operation may be deferred till morning. 

To find the angle which the line as above determined 
makes with the meridian of the point of observation, 
requires a trigonometrical computation. 



Fig. 4. 

observation, and AE, the line of the two stakes. 


Let A be the point of ob¬ 
servation, Z, the zenith of 
that point, HO , an arc of the 
northern horizon, N, the 
north point of that arc, S, 
the north star at its eastern 
elongation, PS, the polar 
distance of the star, AN, the 
meridian of the point of 


The angle sought is NAE — angle PZS = arc NE. 


Now, in the spherical triangle PZS, PZ is the co-latitude 
of the point A, which must be known. Solving this tri- 

sin PS sin polar dist. 

angle, we have sin Z — -, or sin Z —- 

sin ZP cos lat. 

From this, the angle Z becomes known, and, accord¬ 
ingly, it may be formed on the west side of the line 
AE, and thus the direction of the meridian AN deter¬ 
mined. 

On AN, thus found, let a substantial stake be set a hun¬ 
dred yards or more from A, and we have a permanent 
meridian with which we may compare the magnetic 
meridian at any time, and thus determine the declination 
of the needle. 








32 


A MANUAL OF LAND SURVEYING. 


The declination of the needle is the angle which the 
magnetic meridian makes with the astronomical merid¬ 
ian. 

For the purpose, simply, of finding the declination of 
the needle, it is sufficient to lay out on the ground the 
line of direction of the star at one of its elongations, and 
then, knowing the bearing of this line as shown by the 
needle, and the corresponding azimuth of the star, the 
declination of the needle is readily computed. 

Thus, let zb a = azimuth, zb 6 = bearing, and zb d 
= declination, accordingly as they are east or west. 

Then zb d = zb a — (zb 6). 

Rule. —Subtract the bearing f rom the azimuth. 

In applying the Rule, due regard is to be had to the 
algebraic signs. 

A near approximation to a true meridian may be had 

* by observing the pole star while it is 
y-Zeta.. in the same vertical plane with the 
JBear. star Delta, in the constellation Cas¬ 
siopeia. When both are behind the 
plumb-line together, they are very 
nearly in the true meridian. When 
Delta Cassiopeia passes the meridian 
above the pole, it is too high in the 

p heavens to serve this purpose. It 

passes the meridian below the pole 
at midnight April 10th, and may be 
used for two months before and 
after that date. Six months later 
the star Zeta, the last but one in 
the tail of the Great Bear, takes its 

* * place. Fig. 5 shows the relative po- 
sition of these stars and the pole. 


*■ 


Great 


Tfordi 

Pd 


JJeJn 
C as si op 


Fig. 5 







THE TRANSIT 


33 



4 


Fig. 6 









































































































































A MANUAL OF LAND SURVEYING. 





VII. The essential parts of the 
Transit, as shown in the cut, 
are the telescope with its axis 
and two supports, the circular 
plates with their attachments, 
the sockets upon w r hich the 
plates revolve, the leveling 
head, and the tripod on which 
the whole instrument stands. 

The telescope is from ten to 
eleven inches long, firmly se¬ 
cured to an axis having its 
bearings nicely fitted in the 
standards, and thus enabling 
the telescope to be moved in 
either direction, or turned com¬ 
pletely around if desired. 

The different parts of the 
telescope are shown in Fig. 7. 

The object-glass, composed 
of two lenses, so as to show 
objects without color or dis¬ 
tortion, is placed at the end of 
a slide having two bearings, 
one at the end of the outer 
tube, the other in the ring CC, 
suspended within the tube by 
four screws, only two of which 
are shown in the cut. 

The object-glass is carried 
out or in by a pinion working 
in a rack attached to the slide, 
and thus adjusted to objects 
either near or remote as de¬ 
sired. 

The eye-piece is made up 
of four piano convex Knses, 
which, beginning at the eye- 
end, are called respectively the 




































































THE TRANSIT. 


35 


eye, the field, the amplifying, and the object lenses, the 
whole forming’ a compound microscope having its focus 
in the plane of the cross-wire ring BB. 

The eye-piece is brought to its proper focus usually by 
turning its milled end, the spiral movement within 
carrying the eye-tube out or in as desired; sometimes a 
pinion, like that which focuses the object-glass, is em¬ 
ployed for the same purpose. 

1. The Cross -Wires, 

(Fig. 8 ), are two fibres of 
spider-web or very fine plat¬ 
inum wire, cemented into 
the cuts on the surface of 
a metal ring, at right angles 
to each other, so as to divide 
the open space in the center 
into quadrants. 

2. Optical Axis.— The 

intersection of the wires Fio. 8 

forms a very minute point, which, when they are adjusted, 

determines the optical axis of the telescope, and enables 

the surveyor to fix it upon an object with the greatest 

precision. 

The imaginary line passing through the optical axis of 
the telescope, is termed the Line of Collimation, and the 
operation of bringing the intersection of the wires into 
the optical axis is called Adjusting the Line of Col¬ 
limation. This will be hereafter described. 

3. The Vertical Circle firmly secured to the axis of 
the telescope is \\ inches diameter, plated with silver, 
divided to half degrees, and with its vernier enables the 
surveyor to obtain vertical angles to single minutes. 

4. The Level on Telescope consists of a brass tube 
about 6£ inches long, each end of which is held between 
two capstan-nuts connected with a screw or stem attached 
to the under side of the telescope tube. 







36 


A MANUAL OF LAND SURVEYING. 


5. The Magnetic Needle is four to five inches long 
in the different sizes of transits, its brass cup having in¬ 
serted in it a little socket or center of hardened steel, 
perfectly polished, and this resting upon the hardened 
and polished point of the center-pin, allows the needle 
to play freely in a horizontal direction, and thus take its 
direction in the magnetic meridian. The needle has its 
north end designated by a scallop or other mark, and on 
its south end has a coil of fine brass wire, easily moved, 
so as to bring both ends of the needle to the same level. 
The needle is lifted from the pin by a concealed spring 
underneath the upper plate, actuated by a screw shown 
above, thus raising the button so as to check the vibra¬ 
tions of the needle, or bring it up against the glass when 
not in use, to avoid the unnecessary wear of the pivot. 

6. The Lower Plate, called the Limb, is divided on 
its upper surface—usually into degrees and half-degrees— 
and figured in two rows, viz., from 0 to 360, and from 0 to 
90 each way; sometimes but a single series is used, and 
then the figures run from 0 to 360 or from 0 to 180 on 
each side. 

7- The Verniers, of which there are two placed op¬ 
posite each other against the limb, are auxiliary scales 
used in measuring smaller portions of the limb than are 
shown by its graduations. Thirty divisions on the ver¬ 
nier correspond precisely with twenty-nine half degrees 
on the limb. Hence one division on the limb exceeds ( ne 
division on the vernier by one-thirtieth of one-half of a 
a degree, that is, by one minute. 

Accordingly, the number of any division of tho vernier, 
on the side toward which the vernier is moved , which co¬ 
incides with a division of the limb is tho number of 
minutes of arc intercepted by the zero of the vernier and 
the last preceding division of the limb. 

Thus, by the device of a vernier we are enabled to 
measure angles to within one minute, although the limb 
of the transit is graduated only to half-degrees. 





THE TRANSIT. 37 

Adjustments.—The principal adjustments of the Tran¬ 
sit are— 

(1) The Levels. 

(2) The Line of Collimation. 

(3) The Standards. 

8. To Adjust the Levels. —Set up the instrument 
upon its tripod as nearly level as may be, and having un¬ 
damped the plates, bring the two levels above and on a 
line with the two pairs of leveling screws; then with the 
thumb and first finger of each hand clasp the heads of 
two opposite, and, turning both thumbs in or out, as 
may be needed, bring the bubble of the level directly 
over the screws, exactly to the centre of the opening. 
Without moving the instrument proceed in the same 
manner to bring the other bubble to its centre; after 
doing this, the level first corrected may be thrown a little 
out; bring it in again; and when both are in place, turn 
the instrument half-way around; if the bubbles both 
come to the centre, they would need no correction, but if 
not, with the adjusting pin turn the small screws at the 
end of the levels until the bubbles are moved over half 
the error; then bring the bubbles again into the centre by 
the leveling screws, and repeat the operation until the 
bubbles will remain in the center during a complete rev¬ 
olution of the instrument, and the adjustment will be 
correct. 

9. To Adjust the Line of Collimation.— To make 
this adjustment—which is, in other words, to bring the 
intersection of the wires into the optical axis of the tel¬ 
escope, so that the instrument, when placed in the middle 
of a straight line, will, by the revolution of the telescope, 
cut its extremities—proceed as follows: 

Set the instrument firmly on the ground and level it 
carefully; and then having brought the wires into the 
locus of the eye-piece, adjust the object-glass on some 
well-defined point, as the edge of a chimney or other 
object, at a distance of from two hundred to five hundred 



A MANUAL OP LAND SURVEYING. 


88 


feet; determine if the vertical wire is plumb, by clamping 
the instrument firmly and applying the wire to the verti¬ 
cal edge of a building, or observing if it will move par¬ 
allel to a point taken a little to one side; should any dev¬ 
iation be manifested, loosen the cross-wire screws, and by 
the pressure of hand on the head outside the tube, move 
the ring around until the error is corrected. 

The wires being thus made respectively horizontal and 
vertical, fix their point of intersection on the object 
selected; clamp the instrument to the spindle, and having 
revolved the telescope, find or place some good object in 
the opposite direction, and at about the same distance 
from the instrument as the first object assumed. 

(treat care should always be taken in turning the teles¬ 
cope, that the position of the instrument upon the spindle 
is not in the slightest degree disturbed. 

Now, having found or placed an object which the ver¬ 
tical wire bisects, unclamp the instrument, turn it half 
way around, and direct the telescope to the first object 
selected; having bisected this with the wires, again clamp 
the instrument, revolve the telescope, and note if the ver¬ 
tical wire bisects the second object observed. 

Should this happen, it will indicate that the wires are 
in adjustment, and the points bisected are with that of 
the centre of the instrument, in the same straight line. 

If not, however, the space which separate the wires 
from the second point observed, will be double the devia¬ 
tion of that point from a true straight line, which may 
be conceived as drawn through the first point and the 
centre of the instrument, since the error is the result of 


1 





two observations, made with the wires when they are out 
of the optical axis of the telescope. 




the transit. 


39 


For, as in the diagram, let A represent the centre of the 
instrument, and BC the imaginary straight line, upon the 

extremities of which the line of collimation is to be ad¬ 
justed. 

B represents the object first selected, and D the point 
which the wires bisected, when the telescope was made 
to revolve. 

When the instrument is turned half around, and the 
telescope again directed to B, and once more revolved, the 
wires will bisect an object, E, situated as far to one side 
of the true line as the point D is on the other side. 

The space, DE, is therefore the sum of two deviations 
of the wires from a true straight line, and the error is 
made very apparent. 

In order to correct it, use the two capstan head screws 
on the sides of the telescope, these being the ones which 
affect the position of the vertical wire. 

Remember that the eye-piece inverts the position of 
the wires, and therefore that in loosening one of the 
screws and tightening the other on the opposite side, the 
operator must proceed as if to increase the error observed. 
Having in this manner moved back the vertical wire 
until, by estimation, one-quarter of the space, DE, has 
been passed over, return the instrument to the point B, 
revolve the telescope, and if the correction has been care¬ 
fully made, the wires will now bisect a point, C situated 
midway between D and E, and in the prolongation of the 
imaginary line, passing through the point B and the cen¬ 
tre of the instrument. 

To ascertain if such is the case, turn the instrument 
half around, fix the telescope upon B, clamp to the spin¬ 
dle, and again revolve the telescope toward C. If the 
wires again bisect it, it will prove that they are in adjust¬ 
ment, and that the points, B, A, C , all lie in the same 
straight line. 

Should the vertical wire strike to one side of C , the 
error must be corrected precisely as above described, until 
it is entirely removed. 


40 


A MANUAL OF LAND SURVEYING. 


10. To Adjust the Standards.—In order that the 
wires may trace a vertical line as the telescope is moved 
up or down, it is necessary that both the standards of the 
telescope should be of precisely the same height. 

To ascertain this and make the correction, if needed, 
proceed as follows: 

Having the line of collimation previously adjusted, set 
up the instrument in a position where points of observa¬ 
tion, such as the point and base of a lofty spire, can be 
selected, giving a long range in a vertical direction. 

Level the instrument, fix the wires on the top of the 
object and clamp to the spindle; then bring the telescope 
down, until the wires bisect some good point, either found 
or marked at the base; turn the instrument half around, 
fix the wires on the lower point, clamp to the spindle, and 
raise the telescope to the highest object. 

If the wires bisect it, the vertical adjustment is effected; 
if they are throwm to either side this would prove that 
the standard opposite that side was the highest, the ap¬ 
parent error being double that actually due to this cause. 

To correct it, one of the bearings of the axis i$ made 
movable, so that by turning a screw' underneath the slid¬ 
ing-piece, as w r ell as the screws w r hich hold on the cap of 
the standard, the adjustment is made with the utmost 
precision. 

11. To Adjust the Vertical Circle.—Having the in¬ 
strument firmly set up and carefully leveled, bring into 
line the zeros of the circle and vernier, and with the tel¬ 
escope find or place some well-defined point or line, from 
one hundred to five hundred feet distant, which is cut by 
the horizontal wire. 

Turn the instrument half w r ay around, revolve the tel¬ 
escope, and fixing the wire upon the same point as before, 
note if the zeros are again in line. 

If not, loosen the capstan-head screws w r hich fasten the 
vernier, and move the zero of the vernier over half the 
error; bring the zeros again into coincidence, and proceed 


THE TRANSIT. 41 

precisely as at first, until the error is entirely corrected 
when the adjustment will be complete. 

It is not always convenient to make this adjustment so 
as entirely to eliminate the index error. In this case, the 
error should be noted and the proper correction made in 
measuring a vertical angle. 

To find the index error we have the following 

Rule, — Level the instrument and direct the telescope 
upon some well defined spot. Note the reading of the 
circle. 

Reverse the telescope and, turn the vernier plate 180°. 
Direct the telescope upon the point and note the reading 
of the circle. 

Subtract the first reading from the second, and divide 
the remainder by 2. 

12. To Run a Line with the Transit, 

1. Setting up the Transit. —Set the instrument up over 
the starting point, centreing it by means of the plumb 
line. While doing so, place it as nearly level as possible, 
leaving as little as may be, to be done in leveling up the 
plates by the leveling screws. There is opportunity for 
the display of a good deal of skill in setting up a transit 
over a point, quickly, and in proper position. For hill 
sides, a tripod having adjustable legs, called an extension 
tripod, is a great convenience. When the legs are not 
adjustable, set one leg of the tripod down hill and two 
legs on the upper side of the line. It is important that 
the instrument should stand firmly on the ground. Some 
soils are so yielding that it is impossible for the man at 
the transit to change the weight of his body from one 
foot to the other, without getting the transit out of posi¬ 
tion. One remedy is, to not change the centre of grav¬ 
ity of the person, after the transit is in position, until 
the observation is taken. Another is, to drive stout 
stakes into the ground, to set the transit legs on. An¬ 
other is to make a bridge of planks or poles for the transit- 
man to stand on, so as to carry the bearing of his weight 


42 


A MANUAL OF LAND SURVEYING. 


as far as possible away from the instrument. Sometimes 
the aid of an assistant will need to be called in, so that 
the transitman need not move around the transit before 
sighting. 

When the transit is set up firmly in place, loosen the 
lower clamp and turn the instrument on the spindle till 
the level tubes are each parallel to an opposite pair of 
the leveling screws. 

Turn the parallel pair of screws both inward or out¬ 
ward until the bubble comes to the centre. Each level 
being treated in this way, the limb of the instrument is 
caused to be parallel to the horizon. 

Unclamp the vernier plate and set the zero of the ver¬ 
nier to coincide with the zero of the limb. Clamp the 
plates in this adjustment. The leveling screws should be 
kept bearing equally against the plates. 

Do not turn the leveling screws up too tightly. It 
tends to spring the plate and causes unnecessary wear of 
the screw threads. Simply bring them to a firm bearing. 

2. Assistants and their Duties. The Rod- 
man.— A rodman, often called a flagman, using a rod 
called a color pole, and one or more axemen are needed. 
The color pole is often carried by the head chainman. 

The man who carries the color pole, selects places to 
set up the instrument, and gets the transit points, is a 
very important factor in running a line. Nearly as much 
depends upon him for accuracy and speed as upon the 
transitman. He should be thoroughly drilled in his duty. 
He should hold the color pole perpendicularly, clasping it 
lightly between the thumb and forefinger of both hands, 
and the hands held above the head. The point should be 
lifted a little above the ground or hub. He must keep it 
squarely in front of him, and move his body the same 
distance that he does the color pole, when getting a point. 
As soon as the “ All Right” signal is given, let go of the 
pole. It will fall vertically and make the point plain. 
If the pole is held to one side it is apt to have some 


THE TRAHSTT. 43 

uneven pressure given which will make it incline more 
or less. 

A man cannot stand awkwardly and hold a color pole 
accurately. He must be able to judge of the stability of 
the ground to set up on. He must select places where 
the longest sights can be had, and in running through 
timbered country he should select transit points where 
the ground begins to ascend or descend. If any deep 
ravines or gullies are to be crossed, he must select points 
to get across them with the least possible chopping, and 
without having to set up on a steep hillside. He should 
not select a point on the shaded side of a big tree, but 
where the most light comes in through the leaves. A 
small limb cut out of the way will often let in a wonder¬ 
ful amount of light, or a white handkerchief spread over 
the chest, or a light colored straw hat held in the right 
position, sometimes reflects enough light to show clearly 
objects which before were indistinct. In fact, he must be 
a man of gumption and equal to any emergency. But he 
cannot do good work unless he is provided with a good 
color pole. 

3. The Color Pole. —It should be made from a good 
piece of straight grained timber. White or Norway pine 
is good. It is fitted at the bottom with a shoe made from 
gas pipe, with a steel point welded on, and finished by 
turning down in a machine. The shoe ought to be of 
sufficient weight to bring the centre of gravity within 
two feet of the bottom, so that it will have a greater 
tendency to hang vertically when held up. 

The sizes of color poles vary according to the places 
where the> are used. If one is dressed down with planes 
to a six or eight-sided stick, tapering slightly toward the 
top, it will keep straight much longer than a stick turned 
in a lathe. The shoe should be made of sufficient size to 
receive the stick, without dressing it down to go into the 
socket. When finished it should be thoroughly tested, to 
see if the point of the shoe has been set in line with the 



44 


A MANUAL OF LAND SURVEYING. 


centre of the pole. Suspend a plumb bob from a point 
in a ceiling, and mark on the floor the point carried down. 
Fasten a string in the centre of the top of the color pole 
and suspend it from the same point. If the point of the 
shoe covers the mark on the floor it is all right. Prying 
with a color pole should be prohibited. 

4. Axeman. The axemen provide pickets for back¬ 
sights, clear the line of brush and trees, and drive stakes 
and hubs for transit points. They should keep close to 
the line, so that in clearing through woods they do no 
unnecessary cutting. A clear line two feet wide through 
the brush is generally all that is needed. Hubs for transit 
points should be cut square on top and driven firmly into 
the earth, nearly level with the surface. 

5. Projecting the Line. The flagman selects the 
point and, facing the transitman, holds the color pole 
directly in front of him, and guided by the transitman, 
places it in line and makes a mark in the ground. The 
axeman then drives a hub at the place and the rodman 
again holds up his pole and finds the exact point where 
the line crosses the hub and a tack is driven. For most 
surveys a line within the limits of a tack head is consid¬ 
ered close enough. 

The hub for transit point should not be driven near a 
large tree, in soft ground, as a breeze will cause the tree 
to sway so as to move the earth for many feet around it. 
For a backsight it is a good plan to set up a picket, 
pointed at the top, so that the point shall coincide with 
the hole in the eye piece of the telescope. Or it may be 
set far enough from the transit so that the point may be 
aligned by the instrument. The picket should be set so 
firmly in the ground that it will retain its place as long 
as it is needed. A root will sometimes so press against 
a picket as to throw the point out of line after it is set. 
It may be necessary to drive the picket with the axe and 
then insert a wooden point in a cleft in the top of the 
picket. Several such points set up in the line before the 
transit is moved help to secure accuracy in the line. 


THE TRANSIT. 


45 


When the backsight is set, the transit is taken forward 
and set up over the tack point in the hub. The lower 
clamp is loosened, the telescope reversed and sighted to 
the backsight and the instrument clamped in that posi¬ 
tion. The telescope is then righted and the line contin¬ 
ued to the next tack point. When two or more backsight 
points are visible at once, any error in the adjustment of 
the instrument or in running the line will be readily 
detected, and the proper correction may be applied. 

If the line of collimation is out of adjustment and it is 
not desirable to stop and adjust it, the lower clamp is 
loosened, the instrument turned half way round and 
clamped on the backsight. The telescope is then reversed 
on its axis and a second point marked beside the first. 
(See Fig. 9.) A tack is then driven in the true line, which 
is midway between the two. If the instrument is much 
out of adjustment it may be necessary to drive three 
hubs for this purpose. The transit is then set up in the 
true line, and the line continued as far as necessary, in 
the same manner. Obstacles in line are passed by offsets 
to parallel lines, in the same manner as when running 
lines by pickets or compass. Other methods will be con¬ 
sidered in connection with Angular Measurements. 

Examples, to be solved by the student in the field: 

1. Ilun a line half a mile and mark four or more 
points along the line with hubs and tacks. 

2. Retrace it in the opposite direction, testing the 
points to see how they agree. 

3. Bun a line over a hill, marking points at the top 
and bottom and along the slopes. 

4. Retrace it in the opposite direction, testing the 
points. 

5. Bun a lino across a valley, marking points, and re¬ 
trace it in the opposite direction, testing the points. 


46 


A MANUAL OF LAND SURVEYING 


CHAPTER III. 

Description of Instruments, Continued 



■//jm 




Fig. 10. THE SOLAR COMPASS, 












































THE SOLAR COMPASS. 


47 


1. This instrument, for readily determining a true 
meridian, or north and south line, was invented by 
William A. Burt and John Mullett, of Michigan, and 
patented by Burt in 1836. It has since come into general 
use in the surveys of United States public lands, the prin¬ 
cipal lines of which are required to be run with reference 
to the true meridian. 

The arrangement of its sockets and plates is similar to 
that of the surveyor’s transit, except that the sight vanes 
are attached to the under plate or limb, and this revolves 
around the upper or vernier plate on which the solar ap¬ 
paratus is placed. 

The limb is divided to half degrees, is figured in two 
rows, as usual, and reads by the two opposite verniers to 
single minutes. 

2. The Solar Apparatus is seen in the place of 
the needle, and in fact operates as its substitute in the 
field. 

It consists mainly of three arcs of circles, by which can 
be set off the latitude of a place, the declination of the 
sun, and the hour of the day. 

These arcs, designated in the cut by the letters a, b, and 
c, are therefore termed the latitude, the declination, 
and the hour arcs, respectively. 

3. The Latitude Arc, a, has its centre of motion 
in two pivots, one of which is seen at d, the other is con¬ 
cealed in the cut. 

It is moved either up or down within a hollow arc, 
seen in the cut, by a tangent-screw at f, and is securely 
fastened in any position by a clamp-screw. 

The latitude arc is graduated to quarter degrees, and 
reads by its vernier, e, to single minutes; it has a range 
of about thirty-five degrees, so as to be adjustable to the 
latitude of any place in the United States. 

4. The Declination Arc, b, is also graduated to 
quarter degrees, and lias a range of about twenty-eight 
degrees. 


48 


A MANUAL OF LAND SURVEYING. 


Its vernier, v, reading to single minutes, is fixed to a 
movable arm, h, having its center of motion at the end of 
the declination arc, at g; the arm is moved over the sur¬ 
face of the declination arc, and its vernier set to any 
reading by turning the head of the tangent-screw, k. It 
is also securely clamped in any position by a screw, con¬ 
cealed in the engraving. 

5. Solar Lenses and Lines.— At each end of the 
arm, h, is a rectangular block of brass, in which is set a 
small convex lens, having its focus on the surface of a 
little silver plate, A, (Fig. 11,) fastened by screws to the 
inside of the opposite block. 

On the surface of the plate are 
marked two sets of lines, intersect- 
A ing each other at right angles; of 
these, bb are termed the hour lines, 
and cc the equatorial lines, as 
having reference respectively to the hour of the day and 
the position of the sun in relation to the equator. 

In Fig. 11 the equatorial lines are those on the lower 
block, parallel to the surface of the hour arc, c; the hour 
lines are of course those at right angles to the first. 

6. Equatorial Sights.— On the top of each of the 
rectangular blocks is seen a little sighting piece, termed 
the equatorial sight, fastened to the block by a small 
milled head-screw, so as to be detached at pleasure. 

They are used, as will be explained hereafter, in adjust¬ 
ing the different parts of the solar apparatus. 

7. The Hour Arc, c, is supported by the two pivots 
of the latitude arc, already spoken of, and is also connected 
with that arc by a curved arm,.as shown in the figure. 

The hour arc has a range of about 120°, is divided to 
half degrees, and figured in two series, designating both 
the hours and the degrees, the middle division being 
marked 12 and 90 on either side of the graduated lines. 

8. The Polar Axis.— Through the center of the hour 
arc passes a hollow socket, p , containing the spindle ot 


© 1 


a 

..-l-Nl 


0 



Fig. li. 












THE SOLAR COMPASS. 


49 


the declination arc, by means of which this arc- can be 
moved from side to side over the surface of the hour arc, 
or turned completely round, as may be required. 

The hour arc is read by the lower edge of the gradu¬ 
ated side of the declination arc. 

The axis of the declination arc, or indeed the whole 
socket p, is appropriately termed the polar axis. 

9. The Adjuster.—Besides the parts shown in the 
cut, there is also an arm used in the adjustment of the 
instrument as described hereafter, but laid aside in the 
box when that is effected. 

The parts above described constitute properly the solar 
apparatus. 

Beside these, however, are seen the needle-box, n, with 
its arc and tangent-screw, t, and the spirit levels, for 
bringing the whole instrument to a horizontal position. 

10. The Needle Box has an arc of about 36° in ex¬ 
tent, divided to half degrees, and figured from the center 
or zero mark on either side. 

The needle, which is made as in other instruments, ex¬ 
cept that the arms are of unequal lengths, is raised or 
lowered by a lever shown in the cut. 

The needle-box is attached by a projecting arm to a 
tangent-screw, t, by which it is moved about its center, 
and its needle set to any variation. 

This variation is also read off by the vernier on the end 
of the projecting arm, reading to three minutes a gradu¬ 
ated arc, attached to the plate of the compass. 

11. The Levels seen with the solar apparatus have 
ground glass vials, and are adjustable at their ends like 
those of other instruments. 

The edge of the circular plate on which the solar work 
is placed, is divided and figured at intervals of ten de¬ 
grees, and numbered, as shown, from 0 to 90 on each side 
of the line of sight. 


50 


A MANUAL OF LAND SURVEYING. 


These graduations are used in connection with a little 
brass pin, seen in the center of the plate, to obtain ap¬ 
proximate bearings of lines, which are not important 
enough to require a close observation. 

12. Tjines of Refraction. —The inside faces of the 
sights are also graduated and figured, to indicate the 
amount of refraction to be allowed when the sun is near 
the horizon. These are not shown in the cut. 

13. Principles of the Solar Compass. —The inter¬ 
val between two equatorial lines, cc, in Pig. 10, as well 
as between the hour lines, bb, is just sufficient to include 
the circular image of the sun as formed by the solar lens 
on the opposite end of the revolving arm, h , Pig. 9. 

When, therefore, the instrument is made perfectly hori¬ 
zontal, the equatorial lines and the opposite lenses.being 
accurately adjusted to each other by a previous operation, 
and the sun’s image brought within the equatorial lines, 
his position in the heavens, with reference to the horizon, 
will be defined with precision. 

Suppose the observation to be made at the time of one 
of the equinoxes; the arm 7i, set at zero on the declina¬ 
tion arc 6, and the polar axis _p, placed exactly parallel to 
the axis of the earth. 

Then the motion of the arm h , if revolved on the 
spindle of the declination arc around the hour circle c, 
will exactly correspond with the motion of the sun in 
the heavens, on the given day and at the place of obser¬ 
vation; so that if the sun’s image were brought between 
the lines cc, in the morning, it would continue in the same 
position, passing neither above nor below the lines, as 
the arm was made to revolve in imitation of the motion 
of the sun about the earth. 

In the morning, as the sun rises from the horizon, the 
arm h will be in a position nearly at right angles to that 
shown in the cut, the lens being turned toward the sun, 


THE SOLAR COMPASS. 


51 


and the silver plate on which his image is thrown directly 
opposite. 

As the sun ascends, the arm must be moved around, 
until when he has reached the meridian, the graduated 
side ol' the declination arc will indicate 12 on the hour 
circle, and tho arm h, the declination arc 6, and the lati¬ 
tude arc a, will be i:i the same plane. 

As the sun declines from the meridian, the arm h must 
be moved in the same direction, until at sunset its posi¬ 
tion will be the exact reverse of that it occupied in the 
morning. 

14. Allowance for Declination. —Let us now sup¬ 
pose the observation made when the sun has passed the 
equinoctial point, and when his position is affected by 
declination. 

By referring to the Almanac, and setting off on the arc 
his declination for the given day and hour, we are still 
able to determine his position with the same certainty as 
if he remained on the equator. 

When the sun’s declination is south, that is, from the 
22d of September to the 20th of March in each year, the 
arc 6 is turned toward the plates of the compass, as 
shown in the engraving, and the solar lens, o, with the 
silver plate opposite, are made use of in the surveys. 

The remainder of the year, the arc is turned from the 
plates, and the other lens and plate employed. 

When the solar compass is accurately adjusted, and its 
plates made perfectly horizontal, the latitude of the place, 
and the declination of the sun for the given day and 
hour, being also set off on the respective arcs, the image 
of the sun cannot be brought between the equatorial lines 
until the polar axis is placed in the plane of the meridian 
of the place y or in a position parallel to the axis of the 
earth. The slightest deviation from this position will 
cause the image to pass above or below the lines, and 
thus discover the error. 


52 


A MANUAL OF LAND SURVEYING. 


We thus, from the position of the sun in the solar sys¬ 
tem, obtain a certain direction absolutely unchangeable, 
from which to run our lines, and measure the horizontal 
angles required. 

This simple principle is not only the basis of the con¬ 
struction of the solar compass, but the sole cause of its 
superiority to the ordinary or magnetic instrument. For 
in a needle instrument, the accuracy of the horizontal 
angles indicated, and therefore of all the observations 
made, depends upon the delicacy of the needle, and the 
constancy with which it assumes a certain direction, 
termed the magnetic meridian. 

The principal causes of error in the needle, briefly 
stated, are the dulling of the pivot, the loss of polarity 
in the needle, the influence of local attraction, and the 
effect of the sun’s rays, producing the diurnal variation. 

From all these imperfections the solar instrument is free. 

The sights and the graduated limb being adjusted to 
the solar apparatus, and the latitude of the place and the 
declination of the sun also set off upon the respective 
arcs, we are able, not only to run the true meridian, or a 
due east and west course, but also to set off the horizontal 
angles with minuteness and accuracy from a direction 
which never changes, and is unaffected by attraction of 
any kind. 

15. Adjustments. —The adjustments of this instru¬ 
ment, with which the surveyor will have to do, are sim¬ 
ple and few in number, and will now be given in order. 

1st. To Adjust the Levels.— Proceed precisely as di¬ 
rected in the account of the other instruments we have 
described, by bringing the bubbles into the centre of the 
tubes by the leveling screws of the tripod, and then re¬ 
versing the instrument upon its spindle, and raising or 
lowering the ends of the tdbes, until th > b ibbles will 
remain in the centre during a complete revolution of the 
instrument. 


THE SOLAR COMPASS. 


53 


2d. To Adjust the Equatorial Lines and Solar 
Lenses. —First detach the arm 7i from the declination 
arc, by withdrawing the screws shown in the cut from 
the ends of the posts of the tangent-screw k, and also 
the clamp-screw, and the conical pivot with its small 
screws by which the arm and declination arc are con¬ 
nected. 

The arm li, being thus removed, attach the adjuster in 
its place by replacing the conical pivot and screws, and 
insert the clamp-screw so as to clamp the adjuster at any 
point on the declination arc. 

Now level the instrument, place the arm h on the ad¬ 
juster, with the same side resting against the surface of 
the declination arc as before it was detached. Turn the 
instrument on its spindle so as to bring the solar lens to 
be adjusted in the direction of the sun, and raise or lower 
the adjuster on the declination arc, until it can be clamped 
in such a position as to bring the sun’s image as near as 
may be between the equatorial lines on the opposite silver 
plate, and bring the image precisely into position by the 
tangent of the latitude arc or the leveling-screws of the 
tripod. Then carefully turn the arm half way over, until 
it rests upon the adjuster by the opposite faces of the 
rectangular blocks, and again observe the position of the 
sun’s image. 

If it remains between the lines as before, the lens and 
plate are in adjustment; if not, loosen the three screws 
which confine the plate to the block, and move the plate 
under their heads, until one-half the error in the position 
of the sun’s image is removed. 

Again bring the image between the lines, and repeat 
the operation until it will remain in the same situation, 
in both positions of the arm, when the adjustment will 
be completed. 

To adjust the other lens and plate, reverse the arm, end 
for end, on the adjuster, and proceed precisely as in the 
former case, until the same result is attained. 


"1 


A^iVIAKUAL OF LANL> iSUKV-UXIWO. 


In tightening the screws over the silver plate, care 
must be taken not to move the plate. 

This adjustment now being complete, the adjuster 
should be removed, and the arm h, with its attachments, 
replaced as before. 

3d. To Adjust the Vernier of the Declination Arc. 

—Having leveled the instrument, and turned its lens in 
the direction of the sun, clamp to the spindle, and set the 
vernier v, of the declination arc, at zero, by means of the 
tangent-screw at k, and clamp to the arc. 

See that the spindle moves easily and yet truly in the 
socket, or polar axis, and raise or lower the latitude arc 
by turning the tangent-screw /, until the'sun’s image is 
brought between the equatorial lines on one of the plates. 
Clamp the latitude arc by the screw, and bring the image 
precisely into position by the leveling-screws of the tripod 
or socket, and without disturbing the instrument, care¬ 
fully revolve the arm h, until the opposite lens and plate 
are brought in the direction of the sun, and note if the 
sun’s image comes between the lines as before. 

If it does, there is no index error of the declination arc; 
if not, with the tangent-screw k, move the arm until the 
sun’s image passes over half the error; again bring the 
image between the lines, and repeat the operation as 
before, until the image will occupy the same position on 
both plates. 

We shall now find, however, that the zero marks on the 
arc and the vernier do not correspond, and to remedy this 
error, the little flat-head screws above the vernier must be 
loosened until it can be moved so as to make the zeros 
coincide, when the operation will be completed. 

4th. To Adjust the Solar Apparatus to the Compass 
Sights.— First level the instrument, and with the clamp 
and tangent-screws set the main plate at 90° by the ver¬ 
niers and horizontal limb. Then remove the clamp-screw 
and raise the latitude arc until the polar axis is by esti- 






THE SOLAR COMPASS. 


55 


mation very nearly horizontal, and if necessary, tighten 
the screws on the pivots of the arc, so as to retain it in 
this position. 

Fix the vernier of the declination arc at zero, and direct 
the equatorial sights to some distant and well marked 
object, and observe the same through the compass sights. 
If the same object is seen through both, and the verniers 
read to 90° on the limb, the adjustment is complete; if 
not, the correction must be made by moving the sights or 
changing the position of the verniers. 

16. To Use the Solar Compass—Before this instru¬ 
ment can be used at any given place, it is necessary to set 
off upon its arcs both the declination of the sun as affected 
by its refraction for the given day and hour, and the lat¬ 
itude of the place where the observation is made. 

To Set off the Declination.—The declination of the 
sun, given in the ephemeris of the Nautical Almanac 
from year to year, is calculated for apparent noon at 
Greenwich, England. 

To determine it for any other hour at a place in the 
United States, reference must be had, not only to the dif¬ 
ference of time arising from the longitude, but also to the 
change of declination from day to day. 

The longitude of the place, and therefore its difference 
in time, if not given directly in the tables of the Almanac, 
can be ascertained very nearly by reference to that of 
other places given, which are situated on, or very nearly 
on, the same meridian. 

It is the practice of surveyors in the states east of the 
Mississippi, to allow a ilifference of six hours for the dif¬ 
ference in the longitude, calling the declination given in 
the Almanac for 12 m., that of 6 A m., at the place of ob¬ 
servation. 

Beyond the meridian of Santa Fe, the allowance would 
be about seven hours, and in California, Oregon, and Wash¬ 
ington Territory about eight hours. 


56 


A MANUAL OF LAND SURVEYING. 


Having thus the difference of time, we very readily ob¬ 
tain the declination for a certain hour in the morning, 
which would be earlier or later as the longitude was 
greater or less, and the same as that of apparent noon at 
Greenwich on the given day. Thus, suppose the observa¬ 
tion made at a place, say, five hours later than Greenwich, 
then the declination given in the Almanac for the given 
day at noon, affected by the refraction, would be the 
declination at the place of observation for 7 o’clock a.m.; 
this gives us the starting-point. 

To obtain the declination for the other hours of the 
day, take from the Almanac the declination for apparent 
noon of the given day, and, as the declination is increas¬ 
ing or decreasing, add to or subtract from the declination 
of the first hour, the difference for one hour as given in 
the ephemeris, which will give, when affected by the re¬ 
fraction, the declination for the succeeding hour; and 
proceed thus in making a table of the declination for 
every hour of the day. 

17. Refraction.—By reason of the increasing density 
of the atmosphere from its upper regions to the earth’s 
surface, the rays of light from the sun are bent out of 
their course, so as to make his altitude appear greater 
than is actually the case. 

The amount of refraction varies, according to the alti¬ 
tude of the body observed; being 0 when it is in the 
zenith, about one minute when midway from the horizon 
to the zenith, and almost Si' when in the horizon. 

18. Allowance for Refraction—The proper allow¬ 
ance to be made for refraction in setting off the declina¬ 
tion of the sun upon the Solar Compass has long been a 
source of perplexity to the surveyor. Accordingly, a table 
has been prepared, (Table XI), by which the amount of 
refraction for any hour of the day throughout the year 
may be readily obtained. The manner of using the table 
is shown in the solution of the following 


THE SOLAR COMPASS. 


57 


Example. —1. To find the declination for the different 
hours of April 16, 1883, at Troy, N\ Y. 

Solution .—Latitude of Troy, about 42° 30' N. Longi¬ 
tude, 4 hr., 54 min., 40 sec., practically 5 hr. 

Apparent noon at Greenwich is 7 a. m. at Troy. Decli¬ 
nation of sun at Greenwich at noon of April 16, 1883, as 
given by [Nautical Almanac, N. 10° 6' 2"+, and hourly 
change, 53''. 

Refraction in Lat. 42° 30', declination 10°, time 5 hr. 
before noon as given by table, 1' 58''. 

Whence the following figures: 

N. 10° G’ 2'' +Ref. 5hrs. l' 58" = 10° 8' O'' = Dec. at 7 a. m. Troy, 
add lir. dif. 53" 


N.10“ 6 55 ' + 
add hr. dif. 53" 

u 

4 

i t 

l’ ii" 

= 13° 8' O' .6 = 

tt 

8 

N. 10° 7' 48" + 
add hr. dif. 53" 

(( 

3 

it 

0' 52" 

= 10° 8'40" = 

tt 

9 “ 

N.10° 8'41" + 
add hr. dif. 53" 

If 

2 

it 

0' 39" 

= 10° 9' 20" = 

ii 

10 45 

N. 10° 9'34" + 
add hr. dif. 53" 

II 

1 

tt 

O' 36" 

= 10° 10' 10" = 

tt 

11 

N. 10° 10' 27" + 

add hr. dif. 53” 

if 

0 

tt 

O' 36" 

= 13° IP 03" = 

11 

12 M. 

N. 10° IP 20" + 
add hr. dif. 53" 

if 

1 

ft 

O' 36" 

= 10° 11’ 56" 

it 

1 P. M . 

N. 10° 12’ 13" + 
add hr. dif. 53" 

if 

2 

tt 

O' 39" 

= 10° 12’ 52” — 

it 

2 44 

N. 10° 13' 06" + 
add hr. dif 53' 

if 

3 

tt 

0' 52" 

= 10° 13' 58" — 

it 

3 44 

N. 10° 13’ 59" + 
add hr. dif. 53’ 

ii 

4 

it 

1' 11" 

= 10° 15’ 10" - 

11 

4 44 

N. 10° 14' 49" + 

it 

5 

11 

r 58" 

= 10° 16'50" = 

11 

5 44 


Example— 2. To find the declination for the different 
hours of Oct. 16, 1883, at Troy, N. A . 

Solution —Declination of sun at Greenwich at noon of 
Oct. 16, L , as given by Nautical Almanac S.8°51'47".7. 
hourly change 55' 
















58 


A MANUAL OF LAND SURVEYING. 


Refraction 5 hr. before noon, Lat. 42° 30', Dec. — 9% is 
very nearly 9' 24'', and operates to diminish the declina¬ 
tion. 

Whence the following: 

S. 8* 51’ 47 ”.7—Ref. 5 lir. 9’ 24”= 8° 42' 23”= Dec. at 7 a. M. at Troy, 
add hr. diff. 55” 


S. 8° 52’ 42” — 

add hr. diff. 55 ’ 

44 

4 “ 

2’ 49”= 8° 49' 53”= 

44 

8 

S. 8* 53’37” — 
add hr. diff. 55” 

44 

3 “ 

1' 49”= 8” 51' 48”= 

44 

9 

S. 8° 54' 32 ’ — 
add hr. diff. 55” 

*4 

2 “ 

1' 26”= 8° 53’ 06”= 

44 

10 “ 

S. 8° 55'27 — 

add hi-, diff. 55” 

4 4 

1 “ 

1’ 14”= 8° 54' 13”= 

44 

11 

S. 8° 56’ 22” — 

add hr. diff. 55 ’ 

44 

0 “ 

1' 14”= 8° 55’ 08”= 

44 

12 M. 

S. 8° 57’17” — 
add hr. diff. 55” 

44 

1 « 

1' 14”= 8° 56' 03”= 

44 

1 P. M. 

S. 8° 58’ 12 ’ — 

add hr. diff. 55'' 

*( 

2 “ 

etc. 

1' 26”= 8° 56' 46”= 

etc. etc. 

44 

2 “ 


19. To Set Off the Latitude.—Find the declination 
of the sun for the given day at noon, at the place of ob¬ 
servation, as just described, and with the tangent-screw 
set it off upon the declination arc, and clamp the arm 
firmly to the arc. 

Observe in the Almanac the equation of time for the 
given day, in order to know about the time the sun will 
reach the meridian. 

Then, about fifteen or twenty minutes before this time, 
set up the instrument, level it carefully, fix the divided 
surface of the declination arc at 12 on the hour circle, and 
turn the instrument upon its spindle until the solar lens 
is brought into the direction of the sun. 

Loosen the clamp-screw of the latitude arc, and with 
the tangent-screw raise or lower this arc until the image 
of the sun is brought precisely between the equatorial 
lines, and turn the instrument from time to time so as to 
keep the image also between the hour lines on the plate 













THE SOLAR COMPASS. 


59 


As the sun ascends, its image will move below the lines, 
and the arc must be moved to follow it. Continue thus, 
keeping it between the two sets of lines until its image 
begins to pass above the equatorial lines, which is also 
the moment of its passing the meridian. 

Now read off the vernier of the arc, and we have the 
latitude of the place, which is always to be set off on the 
arc when the compass is used at the given place. 

It is the practice of surveyors using the solar compass 
to set off, in the manner just described, the latitude of 
the point where the survey begins, and to repeat the ob¬ 
servation and correction of the latitude arc every day 
when the weather is favorable, there being also nearly an 
hour at mid-day when the sun is so near the meridian as 
not to give the direction of lines with the certainty re¬ 
quired. 

20. To Run Lines with the Solar Compass.—Hav¬ 
ing set off in the manner just given, the latitude and 
declination upon their respective arcs, the instrument 
being also in adjustment, the surveyor is ready to run 
lines by the sun. 

To do this, the instrument is set over the station and 
carefully leveled, the plates clamped at zero on the hori¬ 
zontal limb, and the sights directed north and south, the 
direction being given, when unknown, approximately by 
the needle. 

The solar lens is then turned to the sun, and with one 
hand on the instrument, and the other on the revolving 
arm, both are moved from side to side, until the sun’s 
image is made to appear on the silver plate; when by 
carefully continuing the operation, it may be brought 
precisely between the equatorial lines. 

Allowance being now made for refraction, the line of 
sights will indicate the true meridian; the observation 
may now be made, and the flag-man put in position. 



60 


A MANUAL OF LAND SURVEYING. 


When a due east and west line is to be run, the verniers 
of the horizontal limb are set at 90°, and the sun’s image 
kept between the lines as before. 

The solar compass being so constructed that when the 
sun’s image is in position the limb must be clamped at 0 
in order to run a true meridian line, it will be evident 
that the bearing of any line from the meridian may be 
read by the verniers of the limb precisely as in the ordin¬ 
ary magnetic compass, the bearings of lines are read from 
the ends of the needle. 

21. Use of the Needle.—In running lines, the mag¬ 
netic needle is alw r ays kept with the sun ; that is, the 
point of the needle is made to indicate 0 on the arc of the 
compass box, by turning the tangent-screw connected 
with its arm on the opposite side of the plate. By this 
means, the lines can be run by the needle alone in case of 
the temporary disappearance of the sun; but, of course, 
in such cases the surveyor must be sure that no local 
attraction is exerted. 

• 

The variation of the needle, which is noted at every 
station, is read off in degrees and minutes on the arc, by 
the edge of which the vernier of the needle-box moves. 

22. Allowance for the Earth’s Curvature—When 
long lines are run by the solar compass, either by the 
true meridian, or due east and west, allowance must be 
made for the curvature of the earth. 

Thus, in running north or south, the latitude changes 
about one minute for every distance of 92 chains 30 links, 
and the side of a township requires a change on the lati¬ 
tude arc of 5' 12'', the township, of course, being six 
miles square. 

This allowance is of constant use where the surveyor 
fails to get an observation on the sun at noon, and is a 
very close approximation to the truth. 

In running due east and west, as in tracing the stand- 


THE SOLAR COMrASS. 


61 


ard parallels of latitude, the sights are set at 90° on the 
limb, and the line is run at right angles to the meridian. 

If no allowance were made for the earth’s curvature, 
these lines would, if sufficiently produced, reach the 
equator, to which they are constantly tending. 

Of course, in running short lines either east or west, the 
variation from the parallel would be so small as to be of 
no practical importance; but when long sights are taken, 
the correction should be made by taking fore and back 
sights at every station, noting the error on the back sight, 
and setting off one-half of it on the fore sight on the side 
toward the pole. 

23. Tim© of Day by the Sun.—The time of day is 
best ascertained by the solar compass when the sun is on 
the meridian, as at the time of making the observation 
for latitude. 

The time thus given is that of apparent noon, and can 
be reduced to mean time by merely applying the equation 
of time as directed in the Almanac, and adding or sub¬ 
tracting as the sun is slow or fast. 

The time, of course, can also be taken before or after 
noon, by bringing the sun’s image between the hour lines, 
and noticing the position of the divided edge of the re¬ 
volving arm, with reference to the graduations of the 
hour circle, allowing four minutes of time for each de¬ 
gree of the arc, and thus obtaining apparent time, which 
must be corrected by the equation of time as just de- 
described. 

24. Caution as to the False Image.—In using the 
compass upon the sun, if the revolving arm be turned a 
little one side of its proper position, a false or reflected 
image of the sun will appear on the silver plate in nearly 
the same place as that occupied by the true one. It is 
caused by the reflection of the true image from the sur¬ 
face of the arm, and is a fruitful source of error to the 


62 


A MANUAL OF LAND SURVEYING. 


* § 

inexperienced surveyor. It can, however, be readily dis¬ 
tinguished from the real image by being much less bright, 
and not so clearly defined. 

25. Approximate Bearings.—When the bearings of 
lines, such as the course of a stream, or the boundaries of 
a forest, are not desired with the certainly given by the 
verniers and horizontal limb, a rough approximation of 
the angle they make with the true meridian is obtained 
by the divisions on the outside of the circular plate. 

In this operation, a pencil, or thin straight edge of any 
sort, is held perpendicularly against the circular edge of 
the plate, and moved around until it is in range with the 
eye, the brass center-pin, and the object observed. 

The bearing of the line is then read off at the point 
where the pencil is placed. 

Time for Using the Solar Compass.—The solar com¬ 
pass, like the ordinary instrument, can be used at all 
seasons of the year, the most favorable time being, of 
course, in the summer, when the declination is north, and 
the days are long, and more generally fair. 

It is best not to take the sun at morning and evening, 
when it is within half an hour of the horizon, nor, for 
about the same interval, before and after it passes the 
meridian. 


II. THE SOLAR ATTACHMENT. 

1. The Solar Attachment is essentially the solar 
apparatus of Burt placed upon the cross-bar of the or¬ 
dinary transit, the polar axis only being directed above 
instead of below, as in the solar compass. A little circu¬ 
lar disk of an inch and a half diameter, and having a 
short round pivot projecting above its upper surface, is 
first screwed firmly to the axis ofthe telescope. 

Upon this pivot rests the enlarged base of the polar 
axis, which is also firmly connected with the disk by four 



THE SOLAR ATTACHMENT. 63 


capstan-head screws passing from the under side of the 
disk into the base already named. 

These screws serve to adjust the polar axis, us will be 
explained hereafter. 



MIL JU A'O ,1 

Wo tool 10 1(0 I 10 120 l 


.HMlni 

. 1 m i 




Fig 12, 













































64 


A MANUAL OF LAND SURVEYING. 


2. The hour circle surrounding the base of the polar 
axis is easily movable about it, and can be fastened at any 
point desired by two flat-head screws above. It is divided 
to five minutes of time; is figured from I. to XII., and is 
read by a small index fixed to the declination circle, and 
moving with it. 

A hollow cone, or socket, fitting closely to the polar 
axis and made to move snugly upon it, or clamped at any 
point desired by a milled-head screw on top, furnishes by 
its two expanded arms below, a firm support for the dec¬ 
lination arc, which is securely fastened to it by two large 
screws. 

3. The declination arc is of about five inches 
radius, is divided to quarter degrees, and reads by its ver¬ 
nier to single minutes of arc, the divisions of both vernier 
and limb being in the same plane. 

The declination arm has the usual lenses and silver 
plates on the two opposite blocks, made precisely like 
those of the ordinary solar compass, but its vernier is 
oik side the block, and more easily read. 

The declination arm has also a clamp and tangent 
movement, as shown in the cut. The arc of the declina¬ 
tion limb is turned on its axis and one or the other 
solar lens used, as the sun is north or south of the 
equator. 

4. The latitude is set off by means of a large verti¬ 
cal limb having a radius of two and a half inches; the 
arc is divided to thirty minutes, is figured from the centre, 
each way, in two rows, viz. from 0 to 80°, and from 90° to 
10°, the first series being intended for reading vertical 
angles; the last series for setting off the latitude, and is 
read by its vernier to single minutes. 

It has also a clamp-screw inserted near its centre, by 
which it can be set fast to the telescope axis in any de¬ 
sired position. 

The vernier of the vertical limb is made movable by 
the tangent-screw attached, so that its zero and that ol 


THE SOLAR ATTACHMENT. 


65 


the limb are readily made to coincide when, in adjusting 
the limb to the level of the telescope, the arc is clamped 
to the axis. 

The usual tangent movement to the telescope axis 
serves, of course, to bring the vertical limb to the proper 
elevation, as hereafter described. 

A level on the under side of the telescope, with ground 
vial and scale, is indispensable in the use of the Solar 
attachment. 

The divided arcs, vernier, and hour circle are all on 
silver plate, and are thus easily read and preserved from 
tarnishing. 

5. Adjustments.—These pertain to the solar lenses 
and lines, the declination arc, the polar axis and hour arc, 
as follows: 

(1) The solar lenses and lines are adjusted precisely 
like those of the ordinary Solar, the declination arm being 
first detached by removing the clamp and tangent-screws, 
and the conical centre with its two small screws, by which 
the arm is attached to the arc. 

The adjuster, which is a short bar furnished with every 
instrument, is then substituted for the declination arm, 
the conical centre screwed into its place, at one end, and 
the clamp-screw into the other, being inserted through 
the hole left by the removal of the tangent-screw, thus 
securing the adjuster firmly to the arc. 

The arm is then turned to the sun, as described in the 
article on the Solar Compass, and reversed by the opposite 
faces of the blocks upon the adjuster, until the image 
will remain in the centre of the equatorial lines. 

(2) The vernier of the declination arc is adjusted 
by setting the vernier at zero, and then raising or lower¬ 
ing the telescope by the tangent-screw until the sun’s 
image appears exactly between the equatorial lines. 

Having the telescope axis clamped firmly, carefully 
revolve the arm until the image appears on the other 
plate. 

o 


66 


A MANUAL OF LAND SURVEYING 


If precisely between the lines, the adjustment is com¬ 
plete; if not, move the declination arm by its tangent- 
screw, until the image will come precisely between the 
lines on the two opposite plates; clamp the arm and re¬ 
move the index error by loosening two screws that fasten 
the vernier; place the zeros of the vernier and limb in 
exact coincidence, tighten the screws, and the adjustment 
is finished. 

(3) To Adjust the Polar Axis.—First level the instru¬ 
ment carefully by the long level of the telescope, using 
in the operation the tangent movement of the telescope 
axis in connection with the leveling screws of the parallel 
plates until the bubble will remain in the centre during 
a complete revolution of the instrument upon its axis. 

Place the equatorial sights on the top of the blocks as 
closely as is practicable with the distinct view of a distant 
object; and having previously set the declination arm at 
zero, sight through the interval between the equatorial 
sights and the blocks at some definite point or object, the 
declination arm being placed over either pair of the cap¬ 
stan-head screws on the under side of the disk. 

Keeping the declination arm upon the object with one 
hand, with the other turn the instrument half around on 
its axis, and sight upon the same object as before. If the 
sight strikes either above or below, move the two cap¬ 
stan-head screws immediately under the arm, loosening 
one and tightening the other as may be needed until half 
the error is removed. 

Sight again and repeat the operation, if needed, until 
the sight will strike the same object in both positions of 
the instrument, when the adjustment of the axis in one 
direction will be complete. 

Now turn the instrument at right angles, keeping the 
sight still upon the same object as before; if it strikes the 
same point when sighted through, the axis will be truly 
vertical in the second position of the instrument. 


THE SOLAR ATTACHMENT. 


67 


If not, bring the sight upon the same point by the other 
pair of capstan-head screws now under the declination 
arc, reverse as before, and continue the operation until 
the same object will keep in the sight in all positions, 
when the polar axis will be made precisely at right angles 
to the level and to the line of collimation. 

It should here be noted that, as this is by far the most 
delicate and important adjustment of the solar attach¬ 
ment, it should be made with the greatest care, the bub¬ 
ble kept perfectly in the center and frequently inspected 
in the course of the operation. 

(4) To Adjust the Hour Arc.—Whenever the instru¬ 
ment is set in the meridian, as will be hereafter described, 
the index of the hour arc should read apparent time. 

If not, loosen the two flat-head screws on the top of the 
hour circle, and with the hand turn the circle around 
until it does, fasten the screws again, and the adjustment 
will be complete. 

To obtain mean time, of course the correction of the 
equation for the given day, as given in the Nautical Al¬ 
manac, must always be applied. 

6. To Find the Latitude.—First level the instru¬ 
ment very carefully, using, as before, the level of the 
telescope until the bubble will remain in the center dur¬ 
ing a complete revolution of the instrument, the tangent 
movement of the telescope being used in connection with 
the leveling screws of the parallel plates, and the axis of 
the telescope firmly clamped. 

Next clamp the vertical arc, so that its zero and that of 
its vernier coincide as near as may be, and then bring 
them into exact line by the tangent screw of the vernier. 

Then, having the declination of the sun for 12 o’clock 
of the given day as affected by the meridianal refraction 
carefully set off upon the declination arc, note also the 
equation of time, and fifteen or twenty minutes before 
noon, the telescope being directed to the north, and the 


68 


A MANUAL OF LAND SURVEYING. 


object-end lowered until, by moving the instrument upon 
its spindle and the declination arc from side to side, the 
sun’s image is brought nearly into position between the 
equatorial lines. Now bring the declination arc directly 
in line with the telescope, clamp the axis firmly, and with 
the tangent screw bring the image precisely between the 
lines and keep it there with the tangent screw, raising it 
as long as it runs below the lower equatorial line, or in 
other words, as long as the sun continues to rise in the 
heavens. 

When the sun reaches the meridian, the image will re¬ 
main stationary for an instant and then begin to rise on 
the plate. 

The moment the image ceases to run below is of course 
apparent noon, when the index of the hour arc should 
indicate XII, and the latitude be determined by the read¬ 
ing of the vertical arc. 

It must be remembered, however, that the angle 
through which the polar axis has moved in the operation 
just described is measured from the zenith instead of the 
horizon as in the ordinary solar, so that the angle read on 
the vertical limb is the complement of the latitude. 

The latitude itself is readily found by subtracting this 
angle from 90°; thus, at Troy, the reading of the limb 
being found as above directed to be 47° 16', the latitude 
will be 90° — 47° 16' = 42° 44'. 

It will be noticed that with this apparatus the latitude 
of any place can be most easily ascertained without any 
index error, as in the usual solar compass. 

7* To Run Lines with the Solar Attachment- 

Having set off the complement of the latitude of the 
place on the vertical arc, and the declination for the 
given day and hour, as in the solar, the instrument being 
also carefully leveled by the telescope bubble, set the 
horizontal limb at zero and clamp the plates together, 
loosen the lower clamp so that the transit moves easily 


THE SOLAR ATTACHMENT. 


69 


upon its lower socket, set the instrument approximately 
north and south, the object end of the telescope pointing 
to the north, turn the proper solar lens to the sun, and 
with one hand on the plates and the other on the revolv¬ 
ing arm, move them from side to side until the sun’s 
image is brought between the equatorial lines on the sil¬ 
ver plate. 

The lower clamp of the instrument should now be fast¬ 
ened and any further lateral movement be made by the 
tangent screw of the tripod. The necessary allowance 
being made for refraction, the telescope will be in the 
true meridian, and being unclamped, may be used like the 
sights of the ordinary solar compass, but with far greater 
accuracy and satisfaction in establishing meridian lines. 
Of course when the upper or vernier plate is unclamped 
from the limb, any angle read by the verniers is an angle 
from the meridian, and thus parallels of latitude or any 
other angles from the true meridian may be established 
as with the solar compass. 

The bearing of the needle, when the telescope is on the 
meridian, will also give the declination of the needle at 
the point of observation. 

The declination of the needle being set off, the needle 
kept then at zero, or “ with the sun,” lines may be run 
by the needle alone, when the sun is obscured. 

The sun, however, must ever be regarded as the most 
reliable guide, and should, if possible, be taken at every 
station. 



70 


A MANUAL OF LAND SURVEYING. 




CHAPTEE IV. 

Measurement of Angles. 

1. The instruments already described are used both 
for running lines and for measuring angles. The transit 
is used where the greatest degree of accuracy is required 
and where angles are to be measured within V or less. 

The compass is used when no great degree of accuracy 
is required and the measurement of an angle within 5 / 
is as close as is ordinarily expected. 

Professional Surveyors are provided with the compass 
or transit in some of their various forms. 

Students and others may or may not have them. In 
case of necessity the tape may be used to measure angles, 
and in connection with the picket, sections of the United 
States Survey may be subdivided, irregular fields meas¬ 
ured, and other similar operations performed, with a ra¬ 
pidity and accuracy equal to, if not superior to work done 
with a compass, the picket being used to run the lines 
and the tape to measure both distances and angles. 

2. To Measure Angles with the Tape. 

This is most conveniently done with the aid of tables 
of trigonometrical functions with which the student is 
supposed to be familiar. 

Prob. 1 . To lay off a right angle f rom a point p in a 
given line AB. 







MEASUREMENT OF ANGLES. 


71 


When the sides of a triangle are to each other as 3, 4 
and 5, the angle between the smaller sides is a right 
angle. Hence to lay off a right angle with the tape or 
chain, stick a marking pin at p and then measure along 
the line p m — 3 and stick another pin at m. m Then from 
p as a center with a radius 4 and from m as a center with 
radius 5 strike arcs intersecting at n. Then will mpn 
he the required angle. If the line pn is to be prolonged as 
a picket line, it will be better to range from, if longer 
sides, as 60, 80 and 100 are used. 

This is the most useful of the many methods of laying 
off a right angle with the tape, and can be applied where 
any method can be. The other methods are, for the most 
part, more curious than useful. The following is one of 
the best of them: 

2d Method. Measure along the line in opposite direc¬ 
tions from^p and stick pins in the line at m and m' mak¬ 
ing pm — pm'. Then from m and m' as centres with 
any radius greater than pm strike two arcs Intersecting 
at n. Mpn is the required angle. 



Fig. 14. 

Prob. 2. From a point p in a'given line AB to rim 
a line making any required angle with the line AB. 

1st Method. From p measure p m equal to the cosine 
of the required angle and stick a pin in the line at m. 
Then from m as a centre with a radius equal to the sine 
of the required angle and from p as a centre and radius r 
strike arcs intersecting at n. Then mpn will be the 
required angle and and n will be points in the required 
line. If r — 100 then the lengths of cosine and sine are 
used just as taken from the table of natural sines, only 





72 A MANUAL OF LAND SURVEYING. 

changing the decimal point. Otherwise the tabular 
numbers must first be multiplied by the length adopted 
for r. 



2d Method. In a similar manner we may use the 
natural tangents and secants. From p and in as centres, 
with the secant and tangent of the required angle as 
radii, strike arcs intersecting at n. Secants not given in 
the table may be found from the table of natural sines 

1 

by the formula secant — - 

cosine. 



Example 1 . Lay off, by the use of sines and cosines, 
an angle of 36° 28 / . 

Solution. — Let r = 100 = pn. Then mn — 50.44, 
pm = 80.4. 

Ex. 2. Lay off by the use of tangent and secant, an 
angle of 25° 20'. 

Solution .— Let r = 100 = pm. Then mn — 47.34; 
pn — 110.64. 

Ex 4. Lay off by each method, angles of 48° 20 / , 63° 15', 
26° 32', 8° 40 7 , 18° 23', 37° 06', 82° 45'. 











MEASUREMENT OF ANGLES. 


73 


3d Method. By chords. From the point p as a 
centre, with any radius,— preferably 100, strike an arc 
mx. Find the natural sine of half the angle. Double 
it for the chord. With this distance as radius, from m 
as a centre, strike an arc intersecting the arc mx at n. 
Then p and n are points in the required line and mpn 
the required angle. 



Example 1. Having run the line from the east quarter 
post of section 26 north to the section corner and marked 
it with a sufficient number of pickets, it is required to 
locate the centre line of a highway commencing at the 
quarter post and running north 22t£° west. 

Solution— Measure north in the line from the quarter 
post the full length of the tape = 100, stick a marking pin 
m carefully in line, and strike an arc to the left around the 
quarter post as a centre. Find the sine of half the angle 
and double it. Sine 11° 15' X 2 = .19509 X 2 = .39018 
or correcting the decimal point 39.018. With this dis¬ 
tance as a radius, from m as a centre, locate the inter¬ 
secting point n which is a point in the required line. 

The student should now select a_ level plat of ground, 
mark out a line upon it with pickets and solve the pre¬ 
ceding examples or similar ones, on the ground, each one 
by the several different methods and compare results, 

Also set pickets at the angles of a field of three or more 
sides and measure the sides and angles of the field. 

3. To Measure Angles with the Compass. 

Set the compass up at the intersection of the lines, be¬ 
tween which the angle is to be measured. Put the sights 
in range with one of the lines and note the reading of the 




74 


A MANUAL OF LAND SURVEYING. 

needle. Then put them in range with the other line 
and again note the reading of the needle. Read 
off from the limb, or calculate the number of de¬ 
grees passed over by the needle between the two 
readings. Inland surveying, a line traced out upon the 
ground is termed a course and the angle which the line 
makes with a north and south line is called its bearing or 
course. In compass work the bearings only are taken. 
The angles between the lines of th survey may be com¬ 
puted therefrom if necessary c They are seldom required. 

In reading and writing down the bearings ii is customary 
to state first the direction of the line from which the 
bearing is taken and then the angle to the east or west, ) 
which the course makes with that line, e. g., North 60 
degrees West. South 5 degrees East. Written N. 60° W; 

S. 5° E. 

It is customary in Land Surveying to refer all lines 
to a meridian real or assumed. The cosine of a bearing 
multiplied by the length of its course is called the 

Latitude. 

The sine of the bearing multiplied by the length of the 
course is called the Departure. 

When desirable to find the angles between two lines 
from their bearings, they may be computed as follows: 

Calling N. and S. meridianal letters, we have for the 
angle between two lines from the same station, the fol¬ 
lowing: 

Principles. —1 . When the meridianal letters are alike 
and the others unlike , the angle is the sum of the bearings. 

(2) When the meridianal letters are unlike and the 
others alike , the angle is the supplement of the sum of the 
bearings. 

(3) When both the meridianal and the other letters are 
alike , the angle is the difference of the bearings. 

/ 

(4) When both the meridianal and the other letters are 
unlike, the angle is the supplement of the difference of the 
bearings . 




MEASUREMENT OF ANGLES. 


75 


Observe that the bearings are given in their proper rel¬ 
ative direction with each other and none of them are 
reversed, as S. E. when it should be N. W. 

Examples. 1. The bearings, of two lines are N. 60° W. 
and N. 3° E. What is the angle between them? 

Ans. 63°. 

2. Required the angles between lines having the fol¬ 
lowing bearings: N. 37° E. and 8. 26° E.; N. 87° E. and 
S. 86° W.; S. 15° E. and S. 26° E. Ans. 117°; 179°; 11°. 

3. Stake out a triangle in the field and take the bear¬ 
ings of the sides. 

Find the angles of the triangle and compare the sum 
with 180. 

4. Stake out fields having 4, 5 and 6 sides. Take the 
bearings and find the angles between the sides. 

4. To Correct Courses of Random Lines. 

Case 1st. — Where the line has but one course. 

Random lines as they are usually called are simply 
trial lines run to find the true line between two fixed 
points which are not visible from each other. These 
lines are usually started from one of the points and run 
as nearly in the true direction as can be estimated. If 
the estimate proves correct, and the line strikes the point 
aimed for, the random becomes the true line. If not, the 
perpendicular distance from the line to the point is 
measured, from which the correction for the course may 
be computed. 


c 



Fig. 18 . 

If PC is made perpendicular to AB as is generally 
the case where randoms are run between corners of the 

CP 

United States survey then Tan. CAP— - whence 

AP 





70 


A MANUAL OF LAND SURVEYING. 


the angle CLIP is found, which is the correction to be 
applied to the bearing. 

The angle CAP , when it is quite small, may be found 
by multiplying 57.3° by PC, and dividing by AC. This 
is called the Fifty-seven and three-tenths rule. 
The rule depends upon the fact that for small angles, AP 
differs insensibly from AC, and CP from the arc sub¬ 
tending the angle CAP. 

Whence, angle CL4P:360°::CP:2X3.1416XAP, 

CP 360° CPX5 7.3° CPX57.3° 

or angle CAP — —X-=-:—, or- 

AP 6.2832 AP AC 

The semi-circumference of a circle, with radius AP, is 
3.14159265XAP. 

Whence arc V = 3.14159265 X AP 10800. 

If AP = 1 ch., arc V = 0.00029088 ch. = 0.029088 1. 

If AP — 1 mi. = 80 ch., arc V = 0.029088 1. X 80 
= 2.327 1. == 2}{ 1. 

When angle PAC = V and AP or AC = 1 mi., the 
perpendicular PC, without perceptible error, is 2 % links. 
The Line PC is called the departure of AC, for the dis¬ 
tance AP or AC. 

Taking 2% 1, as the departure of 80 ch. at an angle of 
V, the departure for 40 ch., would be y 2 of 2% 1. = 1| 1. 
= 1 1. + \ of 11. 

For quite small angles, the departure varies directly as 
the angle. Whence, for 40 ch., the following: 

Dep. for V = 11. -f- a of 11.; 

“ “ 2' = 2 1. + of 2 1.; 

“ “ 3' = 3 1. + J of 3 1.; 

and so on, practically true, to 60' or 1°. 

For any other distance, at the same angle, the depar¬ 
ture varies directly as the distance. Accordingly, 

Given minutes of angle, to find links of departure, 
we have the following: 

Rule.—P o the number of minutes , add its one-sixth 
and multiply the sum by the ratio of the distance to 
40 ch. (Good to sixty minutes.) 







MEASUREMENT OF ANGLES. 


77 


On the following: 

General Rule. — Multiply 0.0291 by the number of 
minutes , and multiply the produet by the number of 
chains in the distance. (Good to 240 minutes.) 

Example .—Given angle — 30 r and distance — 23.20 cli., 
to find the departure. 

Since for 40 ch., V of angle gives 1£ 1. of depar¬ 
ture, we may say, without sensible error for a small angle 
that 11. of departure gives f of V of angle, for the same 
distance. 

Or as it may be written, 

Dep. of 11. = V — ^ of V. 

Similarly, “ “ 21. = 2' — \ of 2', 

“ “ 3 1. = 3' — \ of 3', 

and so on, practically true to 60' or 1°. 

For any other distance with the same departure, the 
angle varies inversely as the distance. Accordingly, 

Given links of departure, to find minutes of angle , 
we have the following: 

Rule. — From the number of links of departure , sub¬ 
tract its one-seventh and divide the remainder by the ratio 
of the distance to 40 ch. (Good to 60 minutes.) 

General Rule .—Multiply 0.0291 by the number of 
chains in the distance , and divide the number of links of 
departure by the product. (Good to 240 minutes). 

In the Table of Departures, the-value of PC in chains 
and decimals is given for angles from V to 60', and for 
the distances most commonly required in making resur¬ 
veys and subdivisions of Sections ot the United States 
Survey. To use the Table: Having measured the outing 
PC on the ground, find the nearest tabular number in the 
column for the corresponding distance. 

The angle will be found in the minute column. 

Example 1. Commencing at the west quarter post of 
Section 16, and running north, the random line intersected 


78 A MANUAL OF LAND SURVEYING. 

r> 

the north line of the section, 15 links east of the corner. 
What is the amount of the correction for course ? 

Solution. In 40 chain column, nearest number .151. 
Corresponding number of minutes 13. 

2. Commencing at the south quarter post of section Id 
with declination of needle estimated at 2° 17 r E. set oft' 
on the vernier, ran north on random and intersected the 
north line of the section, 42 links east of the quarter post. 
What is the declination of the needle as referred to the 
quarter line? 

Solution. Distance 80 chains, correction 18C As the 
line came out east of the corner, it is evident that the 
angle between the magnetic meridian and the quarter 
line was 18' greater than was estimated, = 2° 35 r . 

Note.— The North and South lilies of the United States Survey 
are, in a legal sense all true meridians, whatever they may he astron¬ 
omically, and their locations are fixed by the monuments planted for 
the section corners and quarter posts. Hence it is a custom among 
Surveyors to refer the declination of the needle—or the variation as it 
is more frequently called, to these lines, and to mark on each line 
on their plats, the declination for that line. Under that custom the 
line referred to in Example 2 would be marked Var. 2° 35' E. 

3. “East on random between Sections 13 and 24. 

79.98 chains intersected east boundary 34 links south of 
post.” What is the bearing of the corrected line running 
west? Ans. S. 89° 45' W. 

Case 2nd.— Where the line is a broken one of several 
courses. 

Surveyors are frequently called on to retrace the lines of 
angling roads to settle the boundaries of adjacent lands, 
or to locate meander lines, or to find the boundaries of 
irregular tracts, where several courses have to be run 
between the nearest known points of the original survey. 

In such cases random lines are run according to the 
notes of the original survey, and temporary stakes driven 
at the angles of the random line. It will generally be 
found that corrections for course or distance or for both 
will have to be made to place the stakes in their correct 
location. 



MEASUREMENT OF ANGLES. 


79 


Problem. —To correct a random line of several courses. 

In Fig. 19 let A, B, C, i> represent the lines and angles 
of the original survey between the known points A and D. 



Let 1)' represent the terminus of a random run to re¬ 
trace these lines, the direction and distance of which 
from I) is known. 

From A draw the line AD, producing it indefinitely 
beyond 7); also, from A as a centre, with radius AD, draw 
an arc through D. Now, if the error in the random was 
of direction only, then the point D' would be in the arc. 
If it was an error of the chain only, D' would be in the 
line AD or AD produced. Hence the position of D' with 
reference to the arc and the line AD indicates the kind 
of correction and in what direction it is to be applied. 
AD 

-— is the length of the original chain in terms of the 
AD' 

chain used on the random. That portion of the arc 
which is intercepted between the point D and a line 
joining AD', measures the angle of correction. In the 
field we may calculate the course and length, and 
run a sufficient part of the line D'A, and then trace the 
arc from D to its intersection with that line, and thus 
find the relative length of the lines AD and AD', by which 
to determine the correction for the chain and also find 
the chord of the angular correction; or they may be 
calculated as shown in the following example: 

Example 1. — The boundaries of a farm between the 
nearest known monuments are as follows, (See Fig. 19): 

1. N. 16°, E. 12.00 chains. 

2. N. 72°, E. 26.00 “ 

3. S. 22°, E. 14.00 “ 




80 


A MANUAL OF LAND SURVEYING. 


A random was run with var. 2° 30 / E. and came out 
N. 28° E. 32 links from the monument. Required the 
correction for the variation of needle and for the stakes 
in the angles of the random line. 

We will first find the total latitudes and departures of I 
each station on the random line, and the direction and 
distance of a line, AD', which will join the termini. 



N. Lat. 

S. Lat. 

E. Dep. 

Tot. Lat. 

Tot. Dep. 

1. N. 16° E. 12.00 

11.54 


3.31 

11.54 

3.31 

2. N. 72 E. 26.00 

8.03 


24.73 

19.57 

28.04 

3. S. 22 E. 14.00 


12.98 

5,24 

6.59 

33.28 


If we now divide the total departure of the point 7)' 
by its total latitude we will have the tangent of the 
bearing of the line D'A. 

33 28 

— = 5.050 = tan 78° 48 / or S. 78° 48' W. 

6.50 

The length of the line D'A = +6.59 2 + 33.28 2 = 33.927, 

Tf we now subtract the bearing of the line D'D from 
the bearing of the line D'A we shall have the angle 
DD'A = 78° 48 / — 28° = 50° 48'. Let DH be a perpen¬ 
dicular from D to the line AD'; then we have the follow 
ing equations: 

DH — D'D sin AD'D — .32 X .77494 •= ,24798 f. 

D'H = D'D cos AD'D = .32 X • 63203 = . 20225. 

AH = AD' — D'H = 33.926 — . 20225 = 33.7237+. . 1 

DH 

-= tan DAD' = .24798 f -s- 33.7237+ .=■ .00735 = 

AH 

tan 25' = correction for course. 

;_ ah 

AD = i/AH 2 + // /> 2 =-= 33.7237 . 99997 = 

cos DAD' 

33.724. When the angle DAD' is small, AD and AH may 
be considered equal, without sensible error. 4 

AD 33.724 

-=- = .99404 = length of original chain in 

AD' 33.926 • |, 

terms of the chain used on the random. As the random 























MEASUREMENT OF ANGLES. 


81 


came out to the left of the true line the variation, 2° 30' E., 
was too great, hence we subtract the 25 / , giving 2° 05 / 
as the variation of the needle from the meridian of the 
original survey. To find corrections for the stakes it will 
be better to refer them to the meridian of the random, 
hence we will now apply the corrections for course and 
distance to find the courses and distances of the original 
survey, as they would be according to the meridian and 
measure of the random. This done, we calculate their 
total latitudes and departures. The difference between 
these and the latitudes and departures of the correspond¬ 
ing points of the random is the correction to be applied. 



N. Lat. 

S. Lat. 

E. Dep. 

Tot. Lat. 

Tot. Dep. 

1. N. 1G°25'E. 11.928 

11.44 


3.37 

11.44 

3.37 

2. N. 72 25 E. 25.844 

7.81 


24.64 

19.25 

28.01 

3. S. 21 35 E. 13.916 


12.94 

5.12 

6.31 

33.12 


The last course is computed in this table simply as a 
check on the work, as it was a condition of the problem 
that the line DD' was N. 28°, E. 32 links; from which it is 
known that the difference between the two points is: 
latitude 28 Iks., and departure 15 Iks. We will now com¬ 
pare the results in the two tables and find the correction 
at B , C and D. 



Lat. 

B 

Dep. 

Lat. 

C 

Dep. 

Lat. 

D 

Dep. 

Random Line— 

11.54 

3.31 

19.57 

28.04 

6.59 

33.28 

Original Line— 

11.44 

3.37 

19.25 

28.01 

6.31 

33.13 

Correction_ 

S. 10 

E. 6 

S. 32 

W. 3 

S. 28 

W. 15 


Example 2. —Description of a highway between two 
known points: 

1. N. 62° E. 14.00 chains. 

2. N. 4334° E. 8.00 “ 

3. N. 5° W. 12.00 “ 

4. N. 72^° E. 10.25 “ 

5. S. 12° W. 6.43 “ 

A random run with var. 2° 17 r E. came out 62 Iks. east 
of the point. What is the correction for variation of 


























82 


A MANUAL OF LAND SURVEYING. 


needle, and what change must be made in the position of 
each stake at the angles of the random ? 

5. To Measure Angles with the Transit. 

1. Set up the transit at the apex of the angle and set 
the zero of the vernier to coincide with the zero of the 
limb. Clamp the plates in this adjustment and with the 
clamp to the spindle loosened, turn the telescope in the 
direction of one of the lines. Clamp the spindle and 
bring the wire exactly to centre the line by the slow 
motion screw to the spindle clamp. Unclamp the vernier 
and turn the telescope in the direction of the other line. 
Clamp the vernier in that position and make the final i 
adjustment of the wire to the line by the use of the upper 
tangent screw. The angle may then be read from the 
limb. 

2. Instead of first setting the verniers at zero they may 
be clamped in any position on the limb and then the differ¬ 
ence in the two readings will be the angle. When great 
accuracy is required numerous readings of the angle are 
taken on various parts of the limb and the mean of the 
several results taken for the final reading. 

3. To find the angle which the parts of a broken line 
form with any given line. 


Ci 

I 



Fig. 20. 




MEASUREMENT OF ANGLES. 


83 


Suggestions —Let ABODEF be a broken line, and 
suppose it is required to find the angles which the parts 
BC, CD , DE and EF form with the line AB. 

Set the transit at B, with the vernier set at zero. 
Loosen below, reverse the telescope and direct it to A. 
Clamp the limb, revolve the telescope on its horizontal 
axis, unclamp the vernier and direct the telescope to C. 
The reading of the instrument will be the angle bBC 
the line which BC forms with the line AB. 

Remove to C ; and, leaving the vernier clamped, un¬ 
clamp below, reverse the telescope, and direct it to B. 

The limb remaining securely clamped, revolve the 
telescope, unclamp the vernier, and direct to D. The 
reading will now be the angle cCD which the line CD 
forms with the line Cc or its parallel AB. 

The work goes on in this manner to its close. 

Let the student further describe it. 

If the broken line enclose a field, the reading of the 
instrument when set as at A and directed to B , having 
gone entirely around the field, should be 360°. This con¬ 
stitutes a check against errors occuiring anywhere in the 
work. 

4. To measure an angle of elevation or depression. 



Suggestions — Set the instrument at the vertex of 
the angle and level the horizontal limb. 










84 


A MANUAL OF LAN]) SURVEYING. 


Revolve the telescope upward or downward as the 
case may require, and adjust the line of sight to 
the inclined side of the angle. Take the reading of the , 
vertical circle, applying the proper correction for index 
error. 

Otherwise, take the reading of the circle, repeat the 
observation with the telescope and vernier plate reversed, 
and find the mean of the two readings for the angle 
sought. 

6. Verniers are auxiliary scales for measuring 
smaller portions of space than those into which the 
main scale is divided. They are movable beside the 
main scale and are divided into parts which are either 
a little shorter or a little longer than the parts into which 
the main scale is divided. This small difference in 
length is what we are enabled to measure. 

When the limb of a transit is divided to half degrees it 
is common to make either 29 or 31 divisions of the 
Vernier Scale equal to 30 on the limb, making each 
division on the vernier 31' or 29' in length. 

The zero of the Vernier Scale is the point to which the 
reading is to be taken. Suppose the zero line of the vernier 
to make a straight line with some even division of the limb 
and each division on the vernier scale is 29 r in length. 
Now if the Vernier be moved V, the first line of the 
Vernier Scale from zero in the direction in which 
the vernier was moved, will be in a line with the first 
division on the limb. If moved 2 / the second lines 
will coincide; if 3 7 the third lines ; and so on to the 
end of the scale. Such a vernier is called direct reading. 

It is the kind most commonly used on surveyors’ instru¬ 
ments. 

Suppose however that the spaces on the vernier were 
31' long. Then when the vernier was moved forward V 
the first line back of the zero point would coincide with 
the line in the limb and so on. Such a vernier is called a 
retrograde vernier. 







MEASUREMENT OF ANGLES. 


85 


To read any vernier. If the zero of the vernier coin¬ 
cides with any division of the scale, that will be the cor¬ 
rect reading. If not, note the nearest next less division 
on the limb, and then look along the vernier scale till a 
line is found which coincides with a line on the limb. 
The number of this line on the vernier tells that so many 
of the subdivisions which the vernier indicates (usually 
minutes) are to be added to the reading of the entire 
divisions on the limb. 

If several lines appear to coincide equally well, take the 
middle line. 





86 


A MANUAL OF LAND SURVEYING. 





CHAPTER V. 




Passing Obstacles. Measuring Inaccessible 


Distances. 


Having considered the various methods of running 
lines and measuring angles we are now prepared to take 
up some further problems in passing obstacles in the 
line and measuring inaccessible distances. 

These problems may be solved in the field by the use of 
the picket and tape, the compass, or the transit. 

1. To pass an obstacle in the line and measure the 
distance. 

1st, by Parallel Lines. Prom a in the line AB run 
and measure the line ac in any convenient direction, a 
sufficient distance. From c run cd parallel with AB. 



A 


Fig. 22, 


Prom d, run and measure db equal to and parallel with 
ac. Then ab — cd and b is a point in the line AB. 
When running through heavy forests or towns it will 
often be necessary to run several parallel lines before 
returning to the original line. 

2. By 60° Angles. From a run and measure ac 
making the angle Bac ~ 60°. Bun and measure cb — ac 
and the angle acb 60°. Then b is a point in the line 









PASSING OBSTACLES. 


87 


AB and the angle abc — GO 0 , whence the line may be con¬ 
tinued; ab will equal ac. 



2. To Measure Inaccessible Distances. 

Case 1st. When the points are visible from each other 
as over a stream or pond. 



I. By Similar Triangles. 

From a point a in the line AB, required the distance 
ab across the stream. 

At a erect a perpendicular ac to the line AB. From c 
run a perpendicular to cb intersecting AB at d. Measure 
ac and ad. Then as the triangles cad and bed are similar, 

ac 2 

ad : ac = ac : ab, whence ab = —. 

ad 

There are numerous other devices for obtaining the 
distance ab by similar triangles on the ground. Let the 
student work out some of them in the field. 

















88 


A MANUAL OF LAND SURVEYING. 


2. Method by Tangents. 

Erect a perpendic¬ 
ular to AB at a and 
run it a sufficient 
distance ac. Meas- ^ 
ure the angle acb . 

Then ab = ac X tan 
acb. If ac is made 
100 or 1,000, ab may be read directly from the table of 
natural tangents, observing to put the decimal point in 
the proper place. If acb = 45° then ab = ac. 

3. Method by Sines. 

From a run a line 
ac as most conven¬ 
ient. Measure the 
angles acb and cab 
and the side ac. Com¬ 
pute the angle abc. 

Then sin abc : sin acb Fig. 2g. 

ac sin acb 

= ac : ab ab —-. 

sin abc 




4. Method by Cosines. 

From a run a line 
ac to the point c in a 
line perpendicular to 
AB at b. 

Measure the angle ^ 
cab and the line ac. 

Then abbacy, cos 
cab. 



5. Method by Secants. 


Run ac as be¬ 
fore, to a point c 
from which a per¬ 
pendicular to ac 
will strike the 
the point b. Meas¬ 
ure ac and the an- 



Fig. 28. 
















INACCESSIBLE DISTANCES. 89 


gle bac. Then ab = ac X secant bac. If ac = 100 or 1,000 
the distance ab is taken directly from the table. 


6. By 5° 43 / Angle. 

From a lay off the 
angle bac = 5° 43 7 , 
making be perpen¬ 
dicular to ab. Meas¬ 
ure be. Then ab = Fig. 2k 

106c. This method gives results too large by 1.07 in 1,000. 



Case 2nd. Where the points are invisible from each 
other. 


1. If visible and 
accessible from a 
common point c 
outside the line. 

Measure the lines 
ae and be and the 
angle acb. Sub- Fig. 30 . 

tract this angle from 180° and we have the sum of the 
remaining angles of the triangle, to find the difference. 

abe bac abe — bac 

Then ac -f be : ac — be = tan-: tan- 

2 2 

dbc7-\- bac abe — bac 
And —--!-= abe. 

2 2 

abe -f bac abe — bac 
Also-- bac. 

2 2 

ab = «c X cos bac -f 6c X cos abe. 

If a and 6 are inaccessible from c, the sides ac and 6c 
may be measured by any of the preceding methods. 

2. If instead of 
two lines ac and 6c 
we have a broken 
line of any num¬ 
ber of courses, as 
abode f, the bear- 

















90 


A MANUAL OF LAND SURVEYING. 


mgs of which are referred to the line af as a meridian 
—then the algebraic sum of the products of the cosines 
of the several bearings into their respective distances 
will be equal tq af. 

In the United States Surveys distances across lakes 
and bends of large streams are frequently computed from 
the latitudes and departures of the courses around them. 

Examples. —1. In Pig. 24 ac = 100 ad = 27. Required 
ab. Ans. 370.37+ 

2. Same Figure, ae = 250, ad = 96. Required ab. 

Ans. 651.04+ 

3. Fig. 25, ac — 100, angle c = 61° 20 r . Required ab. 

Ans. 182.9. 

4. Same Figure, ac — 250, angle c = 61° 10'. Required 

ab. Ans. 454.1+ 

5. Fig. 26, ac = 500, angle a = 48° 20', angle c = 118° 

KV. Required ab. Ans. 1011.+ 

6. Same Figure, ac — 658, a = 54° 16', c = 88° 32 / . 

Required ab. Ans. 1087.9+ 

7. Fig. 27, ac = 1,000, angle a = 28° 35'. Required ab. 

Ans. 878.12+ 

8. Same Figure, ac — 950, angle a = 18° 56 / . Required 

ab. Ans. 898.6. 

9. Fig. 28, ac = 100, angle a = 76° 40 / . Required ab. 

Ans. 433.6+ 

10. Same Figure, ac = 250, angle a = 56° 20'. Required 

ab. Ans. 450.97. 

11. Fig. 29, ac = 900, be — 648, angle c — 112°. Re¬ 
quired ab. Ans. 1291. 

12. Given the following courses and distances along a 
broken line between the points a and b. Required the 
distance ab. 

1. N. 18° E. 6.25 chains. 

2. N. 40° E. 8.00 “ 

3. N. 5° W. 12.00 “ 

4. K. 44° W. 8.68 “ 


Ans. 30.26+ chains. 


INACCESSIBLE DISTANCES. 


91 


i3. The field notes of the meanders of a lake in sec¬ 
tions 11 and 12 in the township 1, south, range 10 west, 
meridian of Michigan, —- by the government survey, read 
as follows: 


Courses 

N. 58 E. 
N. 11 W. 
N. 63 W. 

Chs. Lks. 

10.00 

20.00 

5.16 

Began at post in line of sections ll and 12 on 
south side of lake: thence in sec. 12. 

to post in line of sec. ll and 12, N. side of lake. 

N. 63 W. 

5.00 

in section ll. 

S. 60 W. 

6.00 


S. 14 E. 

10.00 


S. 33 W. 

15.00 


S. 51 E. 

10.00 


N.73 l / 2 E 

7.00 

to place of beginning. 


Required the distance between the posts on the oppo¬ 
site sides of the lake. Compute the distance by the mean¬ 
ders on each side of the lake. Compare the results to¬ 
gether, and also with the distance returned in the field 
notes which is 27.27 chains. 


14. There is a cliff beside a railroad in the Wasatch 
Mountains known as the Castle Gate. Desiring to know 
its height above the railroad grade I set up the transit 
at Station 744 of the railroad survey and took the angle 
of elevation to the top of the cliff = 38° 42k Elevation of 
station 744 = 6573.62 ft. 

Height of instrument above station 744 =• 4.84 ft. 

1 next went to station 748 in the line with and 400 ft. 
farther away from the clilf and again took the angle of 
elevation to the top of the cliff = 26° 15k 

Elevation of station 748 = 6567.62 ft. 


Height of instrument above the station, 4.56 ft. 



Required the 
height of the Cas¬ 
tle Gate above the 
station 744 and its 
horizontal dis¬ 
tance. 

Answer. 

Height 501.54. 
Distance 620. 


Fjo. 32. 



















92 


A MANUAL OF LAND SURVEYING. 


15. On Christmas 1881 a party of surveyors climbed a 
mountain peak, erected a monument on its summit and 
named it Christmas Peak. Observations from the line 
of the railroad survey were made as follows, the stakes of 
that survey being 100 feet apart: 

From station 933 -f 49.6 P. T. 

Angle of elevation of summit, 23° 42'. 

Angle to right from railroad line ahead, 76° 10C 

Elevation of station, 5005.28 ft. 

Instrument above station, 4.82 ft. 

From station, 940 + 31.4 P. C. 

Angle to left from railroad line back = 82° 18'. In¬ 
quired the height of the peak and its distance from sta¬ 
tion 933 -f 49.6. 


3. Other Methods of Measuring Distances, 

1. To cross a stream or pond. 

Set up the transit at a convenient point, a. Set up a 

rod at b in the line, 
at a convenient dis¬ 
tance, as 100 feet, 
from a. Set up a 
second rod in line at 
e, over the stream* 
Any plain, straight 
rods will answer. Leveling rods with targets are conve¬ 
nient. They should be set up plumb. Mark points d and 
e, in line, on the rods where the horizontal wire of the 
telescope cuts them, liaise or lower the telescope and 
mark two other points,/and g, in line on the rods where 
the wire cuts them. Measure df and eg. Then adf and 
aeg are similar triangles, and df : af :: eg : ag. If df— 1 
and af = 100, eg = 6.25; then ag = 625. 

2. Stadia Measures. 

1. Instead of using two rods as described in the last 
paragraph, two wires are sometimes placed in the dia- 


















INACCESSIBLE DISTANCES. 


93 


phragm of the telescope and adjusted at such a distance 
apart that they will cover a specified space on a rod, as 
1 foot when the rod is 100, 200 or any other specified dis¬ 
tance away. These wires are one on each side of and 
parallel with the horizontal wire of the telescope. They 
may be either fixed on the diaphragm or attached to slides 
by which their distance apart may be adjusted. When 
the wires are adjusted to cover a certain space, as one 
foot on a rod placed 100 feet away, they will cover two 
feet on a rod 200 feet away, or .5 foot on a rod 50 feet 
away. This proportion is strictly true only when the 
measures are taken from a point in front of the in¬ 
strument at a horizontal distance from the object glass 
equal to its focal length. The focal length may be found 
nearly enough by measuring from the plane of the object 
glass to the capstan-headed screws which carry the dia¬ 
phragm. When the telescope is focused on some very 
distant object, as the moon or a star, the horizontal dis¬ 
tance from the plumb line to the point mentioned forms 
a constant which is to be added to all the distances as 
taken from the rod. 

2. It is more convenient, though less accurate, to adjust 
the wires so that they will cover the required space on 
the rod at a specified distance measured from the center 
of the instrument. This method is usually adopted on 
the government surveys, where stadia measures are taken, 
the length of the base being taken at about a mean of 
the distances which the stadia is intended to measure. 
For ail shorter distances the reading is too small. For 
longer distances it is too large. The error is neglected 
as of no consequence in the class of work for which the 
stadia is used. 

When the stadia wires are not adjustable the rod is 
graduated to conform to the wires. A rod is set up at 
the selected distance from the transit. The space inter¬ 
cepted on it by the wires is subdivided decimally, and 
the stadia rod graduated to that scale. 

Where the wires are adjusted to cover a foot on a rod 


94 


A MANUAL OF LAND SURVEYING. 

V 

100 or 200 feet away, the ordinary leveling rod answers 
the purpose of a stadia rod. 

3. In case the measures are not on horizontal lines it 
will be necessary to apply a correction to the stadia read¬ 
ings to reduce them to the horizontal. If the rod has 
been held perpendicular to the line of sight, the horizontal 
distance is found by multiplying the distance to the rod 
by the cosine of the angle of elevation or depression. 

The position of the rod is determined either by a right- 
angle sight applied to the rod, or by the rodman slowly 
moving the top of the rod back and forth until the 
smallest intercept is obtained. On hillsides it will be 
found quite as easy to hold the rod perpendicular to the 
line of sight as to hold it plumb. 

When the rod is held plumb and the base is measured 
from the point in front of the transit the reduction to 
horizontal is made as follows: 

Let/= focal distance of the telescope, 

r — space intercepted on the rod as held vertically, 
s = image of the same intercepted by the stadia 

wires, 

CO' = line of sight at an angle e with the horizon. 

Let A'B' «= r' 
be the intercept 
on the rod as in¬ 
clined at an angle 
e with the vertical; 

r' 

and let b' =/— be 
s 

the corresponding 
base. Let the an¬ 
gle O'CB or O'CA 
= v. We shall 
then have: 

Angle OCB = e -}- v, and angle OCA ~ e — v> whence 
angle OBC = 90°— (e -f v), and angle OAC = 90°— (e — v). 
The angle O'B'B — 90° + v, and angle O'A'A = 90° — v. 










INACCESSIBLE DISTANCES. 


95 


Tn the triangle O'B'B we have 

O'B' sin [90° — (e -f v)] r' cos (e + v) 

-=-or,-=sr-(a) 

O'B sin (90° -f v) 20'B cos v 

In the triangle O'A'A we have 

O' A' sin [90° — (e — ■»)] r' cos (e — v) 

-=-or,-=-(6) 

O'A sin (90° — v) 2 O'A cos v 


Adding (a) and (6), we obtain 

r' r 

-= 2 cos e (c). 

2 O'B X O'A 


Multiplying (a) and ( b ) together, we obtain 

r' r' cos 2 e cos 2 v — sin 2 e sin 2 v 

4 O'B X O'A cos 2 v 


(d) 


Dividing (c) by (cZ), we have, after a little reduction, 
r cos e 

— = --, (e) 

r' cos 2 e — sin 2 e tan 2 v 
which is an expression of the relation sought. 


Cor.—With the wires adjusted to one foot on the rod 
for a base of 100 feet, we should have 

tan v = 0.005 ft., or tan 2 v — 0.000025 ft. 

Thus, tan 2 v = 0, without material error. 

Whence formula ( e ) becomes r' — r cos e. 

To find the distance CO' we have 

r' 

CO' = d' =/- +/+ o = b' +/+ c. 
s 

Whence, CO = d = (b' + / + c ) cos e. 

For vertical rod we have, b' — b cos e. 

Whence, d = b cos 2 e -f (/ + G ) cos e • (/) 

. The height OO' = h = \b sin 2e + (/ + c) sin e. &) 

Example. —Given e = 10° 30 7 , r = 5.36 ft., and f -f c = 
1 ft., to find cZ and 

Solution. —Suppose the wires adjusted to give 1 ft. on 
the rod to the 100 ft., whence b = 536 ft. 















96 


A MANUAL OF LAND SURVEYING. 


Cos e = 0.983 and cos 2 e = 0.9668. 

Whence, d = 536 X 0.9668 -f- 0.98 = 519.18 ft. 

Sin e = 0.182, and \ sin 2e = 0.1792. 

Whence, h ^ 536 X 0.1792 -f 0.18 = 96.23 ft. 

Formula (/) may be put in the form 

d = b cos 2 e + (/+ c) cos 2 e + (/+ c) cos e (1 — cos e). 

Dropping the last term, we have 

d = (b -f / + cj cos 2 e. (h) 

Assuming/'+ c = 1 ft, as a mean value in different 
instruments, the omission of the term (f + c) cos e 
(1 — cos e ) introduces an error for ordinary elevations of 
less than 0.01 ft, in a base of 1000 ft. 

Moreover, the use of formula (7?) operates to diminish 
the very minute error introduced by use of formula (/') 


For slight elevations, as from 1° to 2°, the reduction to 
horizontal may be omitted. For 5° 4F the amount of the 
reduction is about one per cent. The correction for hori¬ 
zontal measurement is sometimes made by omitting to 
add/‘-f c to the base. 











































INACCESSIBLE DISTANCES. 97 

4. The Gradienter is an attachment to the transit 
for fixing grades and determining distances. 

As made by Gurley, it consists of a screw attached to 
the semicircular expanded arm of the ordinary clamp of 
the telescope axis ; the screw is accurately cut to a given 
number of threads, and passing through a nut in one side 
of the arm, presses against a little stud, A, fixed to the 
inside surface of the right-hand standard. 

In the other side of the semicircular arm is inserted a 
hollow cylinder containing a pin actuated by a strong 
spiral spring, the end of the pin pressing against the side 
of the stud opposite that in contact with the screw. 

Near the other end of the screw, and turning with it, 
is a wheel, or micrometer, the rim of which is plated with 
silver, and divided into 100 equal parts. 

A small silver scale, attached to the arm and just above 
the micrometer wheel, is divided into spaces, each of 
which is just equal to one revolution of the screw; so 
that by comparing the edge of the wheel with the di¬ 
visions of the scale, the number of complete revolutions 
of the screw can be easily counted. 

It will be seen that when the clamp is made fast to the 
axis of the clamp-screw, and the gradienter-screw turned, 
it will move the telescope vertically,- precisely like the 
^ngent-screw ordinarily used. 

And as the value of a thread is such that a complete 
revolution of the screw will move the horizontal cross¬ 
wire of the telescope over a space of one foot on a rod 
at a distance of one hundred feet, it is clear that when 
the screw is turned through fifty spaces on the graduated 
head, the wire will pass over fifty one-hundredths, or 
one-half a foot on the rod, and so on in the same propor¬ 
tion. 

In this way, the gradienter can be used in the measure¬ 
ment of distances, precisely like the stadia. 

8 


98 A MANUAL OF LAND SURVEYING. 

Grades can also be established with great facility, as 
follows: Level the instrument; bring the telescope level 
to its centre by the clamp and gradienter screw; move 
the graduated head until its zero is brought to the edge 
of the scale, and then turn off as many spaces on the 
head as there are hundredths of feet to the hundred in 
the grade to be established. • . 1 

Having a transit with gradienter attachment, let the 
student solve the following problems in the field: 

Prob. 1. To find the grade between two points. 

Suggestions. — Set the instrument over one of the 
points, level the plates and the telescope, and bring the 
zero of the screw to the edge of the scale. 

Set the target of the leveling rod at height of instru¬ 
ment. 

With the rod held upon the other point, note the num¬ 
ber of revolutions of the screw required in bringing the 
cross-wire upon the center of the target. That number, 
as so many feet, is the grade. 

Prob. 2. To find the distance between two points. 

Suggestions.— Set up and adjust the parts of the in¬ 
strument as in Prob. 1. On a leveling rod held upon the 
otlur point, note the number of feet covered by one revo¬ 
lution of the screw, and multiply that number by 100. 

If, in order to cover r feet on a rod at a distance of d 
feet, n revolutions of the screw are required, then w r e 
should have: d : 100 :: r : n; whence d — 100 r-r- n. 

Example. —Given n = 2.30 and r = 5 ft., to find d. 

Result, d = 217.39 ft. 

On inclined ground the horizontal sight line may be 
above or below the rod. In such cases, as in stadia 
measurement, a formula of reduction to a horizontal is 
employed, which may be deduced as follows: 




IN ACCESSIBLE DISTANCES. 


99 


Let CO ~ d (Fig. 34), be a horizontal sight line; 

Angle OCO r = e, the elevation of telescope to foot of 
rod; 

Angle O'CB = v, the angle described by n revolutions 
of the screw; 

O'B' = r', the space on a rod perpendicular to CO', 
subtending angle v, and 

OB — r, the corresponding space on a vertical rod. 

We shall then have, [Formula (a)], 
r' sin [90° — (e -j- -y)] cos e cos v — sin e sin v 


v sin (90° -f- v) cos v 

Whence, r' = r (cos e — sin e tan v). 

Let CO' — d'. Then, tan v = — = —. 

d' 100 

100 r' 100 r( n 

-—-] cos e — sin e X — 

n n ( 100 


Whence, d' = 


100 cos e 

or d/ — r\ -sin e\. (1) 

n 


Now, d = d' cos e. 

( 100 ) 

Whence, d — r\ — cos 2 e — 4 sin 2e y. (2) 

In - \ 


Cor. —If n = 1, we have, 

d' = r (100 cos e — sin e), (3) 

and d = r (100 cos 2 e — \ sin 2e), (4) 
in which r is the space on a vertical rod included by one 
revolution of the screw. 


The numbers by which this value of r must be thus 
multiplied for various elevations are given in Table IX. 

Examples. —1. Given e — 15° 20 r , and ?* = 5.42 for one 
revolution of the screw, to find d' and d. 

Solution. —We find in Table IX, 

factor for inclined distance for 15° = 96.33 
“ “ “ “ 15° 30 r -= 96.09 


Difference for 30 7 = 0.24 
whence, “ “ 20 x == 0.16 










100 A MANUAL OF LAND SURVEYING. 

Whence, factor for inclined distance for 15° 20 / = 1*6.17. 

Accordingly, d — 5.42 X 96.17 = 521.24 ft. 

Again, in Table IX we have 

factor for horizontal distance for 15° = 93.05 
“ “ “ “ 15° 30' = 92.59 

Difference for 3CK0.46 
whence, “ “ 2CK — 0.31 

Whence, factor for horizontal dist. for 15° 20' = 92.74. 

Hence, d = 5.42 X 92.74 = 502.65 ft. 

2. Given e = 10.35 rev. to foot of rod, and r = 6.25, to 
find d' and d. 

Suggestion.— From Table X find the angle e, and solve 
as above. 

When c is an angle of depression, the point O' is the upper end of 
the rod. The application of the formula is, however, the same in this 
case as in the one considered. 

Stadia and Gradienter Measurements are • 

found very convenient in solving some of the problems 
in land surveying, but are almost useless in others. They 
save time and trouble in measuring across streams, bogs 
and other places inaccessible to the chain or tape. They 
furnish a quick and easy means of determining how far 
it is to an object, but a slow one of locating points at any 
desired distance, such as setting stakes for a town plat, a 
ditch line, or a railroad. 





PLATTING. 


101 


CHAPTER VI. 

Platting and Computing Areas. 

1. A Plat or Plot is a representation, upon a small 
scale, of the lines of a survey. Platting is simply sur¬ 
veying on paper. The instruments used are analogous to 
those used in the field. 

Lines are marked upon the paper with pencil or pen 
and ink. Generally they will first be drawn lightly in 
pencil; afterward the permanent lines will be inked, and 
all erroneous or superfluous lines erased. Pencils hard 
enough to hold a fine point without breaking are the best 
for this use. 

The right line pen is used for drawing straight lines. 
It is made in various sizes and forms. One of the best is 
shown at h, in Figure 36. 

The scale of equal parts is the counterpart of the chain 
or tape. A great variety of scales are made. One of the 
most useful is the triangular scale (Fig. 36, e ). It has six 
different graduations, all brought to the edge, so that the 
scale may be laid dow 7 n on the paper and the distance 
marked off directly from the scale. The scale in which 
the inch is divided into 10, 20, 30, 40, 50 and 60 equal parts 
is the one most useful to the surveyor. Paper scales are 
made on fine Bristol board, with any graduation desired. 
They are cheap, and as good as any scale as long as they 
last. The student may make his owm scales on paper. 

The protractor (Fig. 36, a) takes the place of the com¬ 
pass or transit. It is simply the whole or part of a grad¬ 
uated circle or limb. Protractors are made in a great 
variety of forms. One of the cheapest and best has the 



ainijTtTnxiiu 



102 


a manual of land surveying. 




Fig. 3 Q. 





















































PLATTING. 


103 


entire circle graduated to quarter degrees. It is made of 
paper, has the middle part cut out, and fine threads or 
wires crossing at the centre of the circle. A paper 
protractor 14 inches in diameter, graduated to quarter 
degrees, costs from 30 to 40 cents. 

Dividers, (Fig. 36,/) are used to space off distances on 
the plat, or transfer distances from the scale to the plat 
or the reverse. When provided with pen or pencil points 
they are used to strike circles and arcs. When they are 
used for the latter purpose they should have a needle 
point on the stationary leg. 

Parallel rulers, as the name indicates, are used in 
drawing parallel lines. When a paper protractor is used 
in platting, it is found convenient to fasten it at some 
point outside the plat and transfer the bearing of the 
lines from the protractor to the plat by means of the 
parallel rule. The best rule for this purpose moves upon 
rollers, (Fig. 36, d.) 

The straight-edge rider and triangle are also used to 
mark parallel lines, as well as to lay off angles. Many 
other articles will be found convenient in platting. A 
drawing board, made of the softest wood, planed smooth 
and true, and thumb-tacks to fasten the paper to the 
board, may almost be considered as necessaries. Neither 
the student nor surveyor needs many instruments for 
platting, but those he has should be perfect in their kind. 
Jt is not deemed necessary at this point to give further 
details of these instruments and their uses, any sugges¬ 
tion which the student may need being left to the teacher 
to make. 


EXERCISES. 

The first seven exercises are the elementary problems of Geometry, 
and are designed to be solved on paper by use of the dividers and 
ruler. 

2. 1. To draw a straight line equal to a given straight 
line. 

2. To make an angle equal to a given tingle. 


104 


A MANUAL OF LAND SURVEYING. 


3. To draw through a given point a line parallel to a 
given line. 

4. To draw through a given point a line perpendicular 
to a given line. Two cases. 

5. To bisect a given line; a given angle. 

6. To construct lines proportional to given lines. 

7. To construct a polygon similar to a given polygon. 

8. Plat the following lines ; 

(1) 8 chains, to scale of 2 chains to the inch. 

(2) 10 chains, to scale of 5 chains to the inch. 

(3) 10 chains, to scale of 4 chains to the inch. 

(4) 17.25 chains, to scale of 3 chains to the inch. 

(5) 25.40 chains, to scale of 4 chains to the inch. 

9. Plat a triangle whose sides are 13.50 ch., 14.25 ch. and 
16.20 ch., on a scale of 5 chains to an inch; on a scale of 3 
chains to an inch. 

10. Plat a rectangle whose adjacent sides are 9.24 ch. 
and 13.78 ch., on a scale of 4 chains to the inch. 

11. Plat a quadrilateral the sides of which are 22.60 ch., 
14.35 ch., 12.20 ch. and 9.80 cli., on a scale of 4 chains to 
the inch, and having one angle of 83° 30'. 

12. Measure the remaining angles and find their sum. 

13. Plat any figure having five equal sides; measure the 
interior angles and find their sum. 

14. Plat a right triangle having a base of 16.25 ch. and 
a perpendicular of 8.60 ch. Find the remaining side and 
angles of the triangle. 

II. Computing Areas. 

In land surveying the areas are computed in triangles 
and quadrangles. If a field has more than four sides, in 
making the computation it is parted off into triangles 
and rectangles or trapezoids, the area of which is com¬ 
puted and their sum taken. 

1. Area of Triangles. 

1. To find the area of a right angled triangle. 

Multiply the base by one half the perpendicular. 


COMPUTING AREAS. 


105 


2. To find the area of an oblique angled triangle. 
Case 1st.— When the sides are given. 

Let A, B. C represent the 
angles, and a , b , c the sides 
opposite them. 

a —(- b -j- c 

Let - = s. Let x — area. 

2 

Then x = fs(s - a) ( s—b) (s — c). 

Case 2nd .—Having two sides and the included angle 

Let a, b be the sides, C the given angle, and x = area. 
From B drop a perpendicular, d, to the side 6. This 
divides the triangle into two right triangles, the area of 
each of which equals its base multiplied by half the 
perpendicular, cl, and the sum of their areas equals the 
sum of their bases multiplied by half the perpendicular; 

bd ab sin C 

that is, x — —. But d = a sin C. Hence, x = -. 

2 2 


B 



Case 3d. — Given two angles and the included side. 

Let A and B be the angles, and c the side given. 

Find C = 180° — (A + B). Find b. 

c sin B be sin A 

Sin C : sin B :: c : b :. b — - x =- 

sin C 2 


Case 4th. —Given two angles and a, side opposite , (A, Ii 

and a.) 

a sin C 

Find C = 180° — (A + B). Find a = -. 

sin A 


a sin B be sin A 

Find b = -. Then x -- 

sin A 2 


2. Areas of Quadrangles. 

Case 1st. —Squares and rectangles. 

Multiply the base by the perpendicular. 












106 


A MANUAL OF LAND SURVEYING. 


Case 2nd.— Trapezoids. A trapezoid is a figure having 
four sides, only two of which are parallel. 



Its area is equal to the half 
sum of the parallel sides, multi¬ 
plied by the perpendicular dis¬ 
tance between them. 


Trapezoid. Fig. 38. 


Case 3rd —Trapeziums have no two sides parallel. 

The area is found by parting 
off into triangles and comput¬ 
ing their areas. 

1. Having the sides and 
Trapezium. Fig. 39 . angles given. 



Let A, B, C, D represent the angles, and a, 6, e, d the 
sides of the trapezium. Let AC be a diagonal dividing 
the trapezium into the triangles ABC and ADC. In each 
of these we have two sides and an included angle given; 

« 6sin.fr cd sin D 
hence, x = -1-. 

2 2 


2. Given the diagonals of a quadrilateral and an angle 
formed by their intersection, to find the area. 



Solution—Let ABCD be the 
quadrilateral, in and n its 
diagonals, and 0 an angle at 
which the diagonals intersect. 


By Case 2nd. under “Area of Triangles,” 
area AOB — \ AO X BO sin 0 
“ AOD = £ AO X DO sin O 
“ DOC = \ COX BO sin O 
“ BOC = | CO X BO sin O. 

Whence, by addition, area ABCD = \ {AO CO) X 
{BO + DO) sin 0, 

mn sin O 


or, area ABCD — 


2 










COMPUTING AREAS. 


107 


Example .—The diagonals of a four-sided held were 
found to measure 18 ch. and 24 ch. Setting a compass at 
their intersection, the bearings of two adjacent corners 
of the held were found to be N. 301° E. and S. 50° E. 
Required the area of the held. 


Solution .—Applying logarithms in the above formula, 


having found 0 = 991°, we have 


TO = 18 

log 

1.255273 

n — 24 

log 

1.380211 

0 = 99 

log sin 

9.91-4003 

2 ar. co. log 

9.698970 

Area = 213.03 

log 

2.328457 

or, Area = 21.303 A. 



3. Given three sides , a, 6, 

d , and the included < 


and D. (See Fig. 39.) 


Let AC = e, be a diagonal. Let the angle BCA = E, 
BAC = F, and CAD= O. In the triangle ABC the sides 
a,, b and angle B are known. In the triangle CAD the 

f side d only is known. It is required to find the side e and 
the angle C. To hnd G: E + F — 180° — B. By trigo- 

tan E + F tan E — F 

nometry, a -f b : a — b ::-:-, by 

2 2 


which we hnd the sum and the difference of the angles E 
E -f F E — F 

and F. -= F, and G = A — F. 

2 2 


To hnd e: Sin F : sin B :: b : e :. e 

ab sin B cd sin D 

x =-1-. 

2 2 


b sin B 

~ • 

sin F 


4. This method of finding the area of a trapezium may 
be applied to polygons of any number of sides, when the 
sides and angles are given. The polygon is divided into 
triangles two less in number than the number of sides 
Each triangle has two sides and the included angle given 
or readily found. 












108 


A MANUAL OF LAND SURVEYING. 




Take for example the irregular polygon of eleven sides 
shown in Fig. 41, which is divided into nine triangles. 


In the triangles A, B, C and 
D two sides and the included 
angle of each are given. From 
6 the remaining sides and angles 
we find two sides and the in 
eluded angle of the triangles 
E and F, and so each triangle 
m turn furnishes the data for computing the adjacent 
triangle, till all are complete. 



i 


3. Offsets.—When it is desired to find the area of 
a field having irregular sides, such as along a stream or •, 
lake, it is well to run a straight line where most conve¬ 
nient to do so, and then run and measure perpendiculars 
to the margin of the field. These are called offsets. They 
divide the space between the straight line and the margin 

of the field into triangles 
and trapezoids, whose ' 
areas may be computed 
separately and the sum 
Fig. 42. taken. 

If the offsets are equidistant the area may be found by 
the following 

Ivule.— From the sum of the offsets , subtract the half 
sum of the extreme ones , and multiply the remainder by 
the common distance between them. 



4. What is the area in acres of the following right 
angled triangles? . 

1. Base = 23.20 ch., perpendicular = 14.60 ch.‘? 

Ans. 16.936 A. 

2. Base = 19.46 ch., perpendicular — 12.18 ch. ? 

What is the area in acres of the following oblique 
angled triangles : (See Fig 37.) 

3. a — 14.26 ch., 6 = 19.40 cli., c = 12.18 ch.? Ans. 8.666 A. 

4. a= 9.43 “ 6 = 11.61 “ c= 8.42 “ 









COMPUTING AREAS. 109 

5. a= 6.23 “ 6 = 14.26 “ C = 22°40 / ? Am. 1.71+ A. 

6. a = 12.20 “ 6 = 20.00 “ C= 36° 15'? 

7. A = 16° 45 r , R = 82° 30', c* = 21.16 ch. V Arw. 6.458+ A. 

8. A = 35°, B = 62° 42', c = 18.20 “ 

9. A = 46°, B = 58° 15', a = 26.50 “ Am. 40.264 A. 

10. A = 37° 20 / , li = 72° 40 / , a= 19.36 “ 

11. A square field is 6.25 chains on a side. Required its 
area. 

12. A square field contains 20 acres. What is the length 

of its sides? Am. 14.142 ch. 

13. What is the area of a rectangle whose sides are 
16.41 and 8.26 chains ? 

14. A rectangular field containing 16 acres measures 

12.50 chains on the base. What is the perpendicular ? 

Am. 12.80 ch. 

15. Commencing on the margin of a river a line was 
run across a bend 20.00 chains to the margin. Commenc¬ 
ing at the end of the second chain, offsets were taken 
every two chains, to the margin of the river, as follows: 
1.61 ch., 2.27 ch., 3.72 ch., 1.96 cli., 4.23 ch., 2.92 ch., 3.26 ch., 

2.50 ch. and 1.25 ch. Required the area between the line 

and the river. Am. 4.744 acres. 

16. Required the area of a field bounded as follows: 
1st. North 17.65 ch. 

2nd. S. 36° 12 / W. 8.20 ch. 

3rd. S. 12° 34 / W. 7.26 “ 

4th. S. 58° 26 / E. 7.53^ “ 

Suggestion.— First: Change bearings into angles 
between the lines and compute as two triangles. 

Second: Take the first line as a base, divide the figure 
into two right angled triangles and a trapezoid, and com¬ 
pute the area. Compare the two methods as to number 
of figures required for the solution. 

17. The sides of a pentagon measure 6.25 chains each. 
What is its area ? 

Suggestion. —Part the figure into three triangles and 
compute. Also part into five isosceles triangles. Com¬ 
pute and compare the two methods. 


110 


A MANUAL OF LAND SURVEYING. 




5. 1. Rectangular Coordinates. — Let XX' and 

YY' be two lines intersect¬ 
ing each other at right 
angles, as at O. 


K 


•Y 

f 


D 

T 


i 

i 

L 

U 


f 


I 


Let Pi, P 2 , P 3 be any 


n 


points in the plane of the 
lines. 


Let P x a u P 2 a 2 , P 3 a 3 be 


perpendiculars from the 
points upon the axis XX' y 
and P x b u P 2 b 2 , P 3 b 3 be 


Fig. 43, 

perpendiculars from the points upon the axis Fl r/ . 


The distances Oa x , Oa 2 , Oa 3 are called Abscissas of 
the points P u P 3 ; and the distances Ob x , Ob 2 , Ob 3 are 
called Ordinates of the points. 


The point O is called the Origin. 

The abscissa and ordinate of a point are together called 
Coordinates of the points. 

Coordinates at right angles with each other are called 

Rectangular Coordinates. 

It is customary to denote abscissas by x and ordinates 
by y, coordinates of different points in connection with 
each other being distinguished by use of subscripts. 

Thus, of the point P 1} the coordinates Oa 1 and Ob x or 
cl 1 P x may be denoted by x x and y x \ of the point P 2 , the 
coordinates Oct 2 and Ob 2 or a 2 P 2 may be denoted by x 2 
and y 2 ; and so on. 

It will be seen that the coordinates of a point afford 
the means of locating it with respect to the axes. 

The use of longitude and latitude in Geography is an illustration. 

By use of the signs -f and —, the coordinates of any 
point in the plane of the axes are readily expressed. 









COMPUTING AREAS. 


Ill 


EXERCISES. 

2.—1. Construct the point of which x = 4 and y — 7. 

2. Given x — — 5 and y — 3, to construct the point. 

3. Given x — — 3 and y — — 6, to construct the 
* point. 

4. Given x = 6 and y — — 4, to construct the point. 

5. Given x = 0, y — 2; x = — 5, y = 0; x — 0, y = 0. 

Required the points. 


3. Application to Area. — Let it be required to 
find the area of a series of trapezoids included between 
perpendiculars from the points of a broken line upon a 

straight line. Suppose 
the straight line, as OX', 
to be an axis of abscis¬ 
sas, and the first perpen- 
„ dicular at the left, as OA, 

„ to be an axis of ordinates. 

Fig. 44. 



Let etc., be the abscissas of the points A, B, 

C, etc., and y lt y 2 , y 3 , etc., the corresponding ordinates. 

Accordingly, the area of the several trapezoids is 

h [ff a ( 2 /i + 2 / 2 ) + (»» — « 2 ) ( V 2 + 2/3) 

+ (a? 4 — a? 8 ) (2/3 + 2/4) H-(a? n * ^ n -i) (2/n^-i + 2 /n)], 

in which n is the number of trapezoids plus one. 

The above formula may be changed to the form 
1 [a ? 2 (2/1 — 2/3) + (2/2 — 2/4) + a ?4 (2/3 — 2/5) 

H-a'n —1 ( 2 /u—2 — 2 /u) + X r\ (2/n— 1 ~h 2 /n)]- (®) 

Whence, for the area included between a straight line, 
as a base, and a broken line whose points are given by 
their coordinates upon the base, we have the following 


Rule. —From each ordinate subtract the second suc¬ 
ceeding one and multiply the remainder by the abscissa 
corresponding to the intervening ordinate. 

Also , multiply the sum of the last two ordinates by the 
last abscissa. 

Divide the algebraic sum of the products by 2. 









112 


A MANUAL OF LAND SURVEYING. 


The above formula and Rule have been deduced independently of 
any supposition as to the relative directions of the parts of the broken 
line. They are therefore true whatever may be the form of the broken 
line. That is, whether any part should be perpendicular to the base, 
either toward or from it, or whether any part should be turned back¬ 
ward respecting the preceding one. 


Suggestion. —Let the student verify the rule in a case, 



for example, like the < 
following, in which BC 
is represented as being 
parallel to the base, CD 
as perpendicular toward 
it, and FG as being 
turned backward from 
FF. 


Find how it would be, if one or more of the ordinates 
were zero; if one or more were negative. 


EXERCISES. 

4.—1. Given y i =- 12, y. z = 12, y. s = 16, y 4 = 8 and v/ 5 = 6, 
also x x — 10, x 2 = 18, x s = 24, x 4 = 30 and x 5 == 20, to find 
area. 

Given the following, to find area: 

(2) (3) (4) 


140 

1000 

000 

950 


1000 

200 

435 

812 

240 

844 


1150 

317 

250 

725 

30G 

530 


828 

420 

200 

500 

G40 

325 


G50 

305 

300 

450 

415 

200 


4G0 

524 

320 

000 

000 

000 


000 

250 


5. As a second example of the application of coor¬ 
dinates in finding area, let there be taken an ordinary 
polygon, as ABCDEF. (Fig. 46.) 

Let x lt x 2 , x z , etc., be the abscissas of the points A, B , 
C, etc., and y x , y z , y 3 , etc., the corresponding ordinates. 















COMPUTING AKEAS. 


113 





Now, since formula (a) is true for any broken line, it 
holds for the case in which the broken line beginning, as 
at A, returns to the same point, forming thus a polygon, 
as ABCDEFA. 

In this case, the last term of («) vanishes, and we have 
as the area a polygon of n sides, 

1 [®i (Z/m— y>) + a ? 2 ( 2 /i— 2/3) + (2/2— 2/4) + (2/3— 2/5) 

-f etc., to n terms]. (b) 

or, factoring with respect to y, we have the form 

— l[ 2 /i (xn—x-i ) + 2/2 Oi— x-t) + 2/3 O2—*4) f 2/4 (^3—^5) 

-f etc., to n terms]. (c) 

Whence, for the area of a polygon whose vertices are 
given by their coordinates, we have the following 

Rule .—From the ordinate of each vertex subtract the 
second succeeding one , and multiply the remainder by the 
abscissa of the intervening vertex; or, from the abscissa 
of each vertex subtract the second succeeding one , and 
multiply the remainder by the ordinate of the intervening 
vertex. 

Divide the sum of the products by 2 
9 



















114 


A MANUAL OF LAND SURVEYING. 


Soh.~F ormulas (ft) and (c) will be seen to be in accordance with 
any situation of the coordinate axes, agreeably with convenience of 
field work. In particular cases, one or more terms will be found to 
disappear. Due attention to algebraic signs is important. 

The formulas are easy to remember, and simple of application. 
With an instrument adapted to laying off right angles, they afford a 
practical means of computing the'contents of irregular tracts. 


EXERCISES- 

6 . Required the area and a plat of a field the coordi¬ 
nates of whose corners are 

x D — x 6 =0, x x =7 ch., x 2 =12| ch., x. A = 18 ch., x±— 15 ch., 
# 5 = 10 ch.; and 

Va = Ve = 6 ch., y x = 12 ch., y 2 = 20 ch., y z = 15 ch., 
2 / 4 = 8 ? ch., 2/5 = 0 ch. „ Area, 16.175 acres. 

Find the area, supposing a different situation of the 
axes. 


7. Given the lengths and bearings of the sides of a 
polygonal field , to find the area. 


Solution. — Let 
ABODE represent 
the field. Let NS de¬ 
note the meridian of 
the most westerly 
station. This line, 
which may be as¬ 
sumed as passing 
through any station 
at pleasure, but more 
conveniently the ex¬ 
treme western or the 
eastern one, is called 
the Principal Me¬ 
ridian. 

To the principal 
meridian let there be 
drawn from the sev¬ 
eral stations the per- 



Fig. 47 . 



















COMPUTING AREAS. 115 

pendiculars Ba , Cd, Dh and Ek, and upon Cd and Dh let 
there be drawn the perpendiculars Bb, Cc and Ee. 

These perpendiculars are, respectively, the bases and 
the altitudes of trapezoids composing a portion of the 
field. 

Now, if from the sum of the areas of the trapezoids 
the sum of the areas of the triangles ABa and AEk be 
subtracted, the remainder will be the area sought. 

That is, clearing of fractions, 

2 X area pol. = (aB + Cd) Bb + (dC -j- Dh) Cc -f- (hD 4 - Eh) 

X Ee — (aB X a A ) — (EkX kA.) 

It is now to be considered how the dimensions of the 
trapezoids and triangles depend upon the lengths and 
bearings of the sides of the field. 

8 . Latitude and Departure.— For convenience of 
description, let it be supposed that a survey of the field 
above represented was made “ with the land on the right,” 
beginning at A. 

In going from A to B , there was made a distance Aa, 
north, and a distance aB , east; in going from B to C there 
was made a distance Bb, south , and a distance bC, east. 
Finally, in going from E to A, there was made a distance 
kA, north, and a distance Ek, west. Distances made north 
are called Northings, and south, Southings; dis¬ 
tances made east are called Eastings, and west, West¬ 
ings. Northings and southings are together called 
Latitudes, and eastings and westings are called 
Departures. 

It will be seen that the length of a course is the hypot¬ 
enuse of a right triangle of which the latitude of the 
course is the side adjacent to the bearings, and the depar¬ 
ture, the side opposite the bearing. Whence, 

Latitude = length of course X cosine of bearing, and 
Departure = length of course X sine of bearing. 

From these fundamental formulas, several others ex- 


116 


A MANUAL OF LAND SURVEYING. 


pressing relations of either of the four quantities to two 
others are easily derived. 

Thus, denoting the latitude by Z, the departure by d, the 
length of course by c, and the bearing by b , is obtained 
the following 

TABLE OF CASES. 




The Traverse Table.— This table, which is given 
with others in the back part of the book, shows the lati¬ 
tude and departure for any bearing to each quarter degree 
for any distance from 1 to 10. For other distances, the 
latitude or departure is found by adding the latitudes or 
the departures of the partial distances, as shown in the 
following 

EXERCISES- 

9.—1. Find the latitude and the departure for a bearing 
of 24°, for a distance of 7 cli.; for a distance of 5 cli.; for 
a distance of 10 ch. 


No. 

Given. 

Required. 

Formulas. 

1 

b, a 

Z, d 

Z = c COS b 

d — c sin b. 




l 


1 2 

b, l 

c, d 

c =- 

d = l tan b. 




cos 6 





d 

d 

3 

b, d 

c, Z 

(J '- 

l • - 




sin 6 

tan b 




Z 


4 

c, Z 

b, d 

cos b = - 

d = -j/ g 2 — Z 2 . 




c 





d 


5 

e, d 

b, l 

sin b =- 

Z = |/ c 2 — eZ 2 . 




c 





d 


6 

Z, d 

0, c 

tan b — - 

c— |/Z 2 -M 2 . 




Z 




































COMPUTING AREAS. 117 

2. Find the latitude and the departure on a bearing of 
37i°, for a distance of 12 ch. 


OPERATIONS. 


For 37i°, distance 10, lat. = 7.9000 dep. 

“ 2, “ = 1.5920 “ 


u a 


6.0529 

1.2106 


n <« 


19 “ 

1 "> 


9.5520; “ 


i .. 


3. Find the latitude and departure on a bearing of 40|° 
for a distance of 17.23 ch. 


OPERATIONS. 


For 40^°, distance 10, lat. = 7.5756, dep. = 6.5276 


u u 

“ 6 

“ = 5.3030, “ 

= 4.5693 

u u 

“ .2 

“ = 0.15151, “ 

= 0.13055 

« << 

“ .03 

“ = 0.022727, “ 

0.019583 

a u 

“ 17.23 

“ = 13.053, 

11.247 


ANOTHER 

FORM OF WORK. 


3eartny. 

Distances. 

Latitudes. 

Departures. 

O 

e-O'-f 

o 

1000 

07576 

06528 


700 

53030 

45693 


20 

15151 

13055 


3 

22727 

19583 


1723 

1305.3237 

1124.7433 


We take the distance in links, and write the latitude and departure 
for tlie first figure of the number, omitting the decimal point; we write 
under them the latitude and departure for the second figure, setting 
them down one place farther toward the right; under them, the lati¬ 
tude and departure for the third figure, setting them one place farther 
toward the right, and so on. 

We then add the separate latitudes and separate departures, and 
point off four figures from the right. The results thus obtained are the 
latitude and departure sought, as expressed in links. 

Notice that bearings from 45° upward are found in the right hand 
column of the table, and the columns of latitude and departure are 
denoted at the foot of the page. Care needs to be taken here to avoid 
mistakes of latitudes for departures and departures for latitudes. 










118 


A MANUAL OF LAND SURVEYING. 


Find the latitudes and departures for the following 
bearings and distances: 

(1) Bearing 52|°, Distance 437. 

(2) Bearing 65£°, Distance 3669. 

(3) Bearing 211°, Distance 2030. 

(4) Bearing 40°, Distance 506. 

(5) Bearing 8H°, Distance 12.34 ch. 

lO. Meridian Distance.—The distance of a station 
or any point from the principal meridian is called its 
Meridian Distance. The meridian distance of a line 

is the meridian distance of its middle point. If the 
meridian passing through the extreme easterly or west¬ 
erly station of a survey around a tract of land be taken ^ 
as a base and perpendiculars be drawn from it to each 
station of the survey, the tract and the space between 
it and the meridian will be divided into triangles and 
trapezoids whose areas are readily computed. 

Beginning with the station through which the me¬ 
ridian passes which we call Sta. 0, then the meridian i 
distance of Sta. 1 will equal the departure of the first 
course. 

The meridian distance of any {station will equal the alge¬ 
braic sum of the departures of all the preceding courses 
up to that point. 

The meridian distance of any course or line will equal the 
half sum of the meridian distances of the stations at 
the two ends of that course or line. 

The area of any triangle or trapezoid thus formed will 
equal the product of the latitude of the line or course 
on which it is based multiplied by the meridian dis¬ 
tance of that line. 

The area of the tract is equal to the sum of the areas of 
all the triangles and trapezoids thus formed minus the 
sum of the areas of those triangles and trapezoids which 
lie outside the lines of the survey. 

The area of the tract is also equal to the difference 
between the sums of those areas found from latitudes 
which are northings and of those where they are 
soul hings. 


I 







COMPUTING AREAS. 


119 


1\e will now apply the foregoing principles to find 
the area of the tracts described in the following Field 
Notes and shown in the figure. On the figure each sta¬ 
tion is numbered to correspond with the field notes and* 
each line is also numbered in its order as run. The sev¬ 
eral triangles and trapezoids formed by perpendiculars 
from the stations to the meridian are lettered in their 
order. 



Station 

Bearing 

Distance 

0 

N. 26F E. 

12.00 ch. 

1 

N. 59° E. 

9. SO “ 

*> 

~ 

S. 66° E. 

19.60 “ 

3 

S. 35° W. 

15.68 “ 

4 

S. 66° W. 

13.12 “ 

5 

N. 46° W. 

14.72 “ 


Finding from the Traverse Table the latitudes and 
departures to the nearest link, we have 


Bearing 261° 

JJist. 

12.00 

Lat. 10.74 N. 

Dep. 5.35 E. 

U 

59° 

U 

9.80 

“ 5.05 N. 

“ 8.40 E. 

U 

66° 

a 

19.60 

“ 7.97 S. 

“ 17.91 E. 

« 

35° 

u 

15.68 

“ 12.85 8. 

“ 8.99 W. 

u 

66° 


13.12 

“ 5.34 8. 

“ 11.99 W. 

u 

40° 

<; 

14.72 

“ 10.23 N. 

“ 10.59 W. 


Obviously, in going entirely around a field there should 
be made the same southing as northing, and the same 
westing as easting. But from unavoidable lack of pre¬ 
cision in the use of instruments, this is practically seldom 
found to have been done, according to the figures used. 
The error, however, can usually be made very small. 
Finding it large, the entire field work should be reviewed. 

It is not ;i settled point among surveyors how great an error of lati¬ 
tude or departure may he allowed without resurveying the lot. Some 
would admit a difference of one link for every three chains in the sum 
of the distances, others for every five chains, and again others would 
require it to he within one link for every ten chains. 


























120 


A MANUAL OF LAND SURVEYING. 


As a check against errors of bearing, a back sight 
should be taken at every station, and the reverse bearing 
compared with the corresponding direct bearing of that 
station. If the two are found to differ considerably, both 
should be reviewed. Let us now see how small an error 
of latitude and of departure we have in the present case. 

Sum of northings = 10.74 + 5-05 + 10.23 = 26.02. 

“ “ southings = 7.97 + 12.85 + 5.34 = 26.16. 

Difference of latitudes = 00.14 = error of latitude. 

_ Sum of eastings = 5.35 + 8.40 + 17.91 = 31.66. 

“ “ westings = 8.99 + 11.99 + 10.59 = 31.57. 

Difference of departures = 00.09 = error of departure. 

The above errors may be considered reasonably small 
for a field of the size of the present one. 

In practice, some of the courses may have been measured over 
rough or uneven ground, and, accordingly, such courses should beai 
a larger proportion of the error. 

Some of the bearings may have been taken with an indistinct 
sight, which would dictate the allotment of more than a proportionate 
amount of the error to them. 

Distances as measured over uneven ground are liable to be too long. 
In such cases, the length of a course may be diminished when sucli 
change would favor the balancing. Similarly, a doubtful bearing may 
be changed, if the error should appear to be attributable to it. 

It is a common mistake to reverse the position of the latitude and 
departure in the columns. If the bearing is greater than 45° the 
departure is greater than the latitude, and it is less when the bearing 
is less than 45°. Scan the columns for such errors. 

11. Balancing.—The next work is to distribute the 
errors among the several courses in proportion to their 
lengths, in accordance with the following 

Principle .—As the mm of the lengths of all the courses 
is to the length of each course , so is the total error to the 
error of that course. 

This operation is called Balancing. 

Applying the above principle, we divide the errors by 
the sum of the lengths of all the courses and multiply the 
quotients by the length of each course, indicating the 
products as positive or negative, accordingly as they are 
to be added or subtracted in making the required correc¬ 
tion. 


COMPUTING AREAS. 


121 


Thus, 00.14-^-84.92=00.00165; and 00.09-^-84.92=00.00106; 
00.00165X12=00.0198 or +00.02; and 00.00106X12=00.01272 
or —00.01, to the nearest link. 

In the same manner, by multiplying the above quotients 
by the lengths of the other courses, the correction for 
them is readily obtained. 

Collecting results thus found, we have the following 

TABLE I. 


Sta. 

Latitude. 

Departure. 

Cor.L 

Cor I) 

Balanced. 


N. 

S. 

E. 

W. 



N. 

S. 

E. 

W. 

l 

10.74 


5.35 


+.02 

—.01 

10.76 


5.34 


2 

5.05 


8.40 


+ .02 

—.01 

5.07 


8.39 


3 


7.97 

17.91 


—.03 

—.02 


7.94 

17.89 


4 


12.85 


8.99 

—.03 

+.02 


12.82 


9.01 

5 


5.34 


11.99 

—.02 

+.01 


5.32 


12.00 

G 

10.23 



10.59 

+.02 

+.02 

10.25 

_ 



10.61 


We next find the Meridian Distance of the several 
stations. 


M. D. of Sta. l=Dep. of Course 1=5.34. 

M. I) of Sta. 2=M. 1). of Sta. 1+Dep. of C. 2 

=5.34+8.39=13.73. 

M. D. of Sta. 3=M. D. of Sta. 2+Dep. of C. 3 

=13.73+17.89=31.62. 

M. D. of Sta. 4=M. D. of Sta. 3—Dep. of C. 4 

=31.62—9.01=22.61. 

M. I). of Sta. 5=M. I). of Sta. 4—Dep. of C. 5 

=22.61—12.00=10.61. 

M. D. of Sta. 0=M. D. of Sta. 5—Dep. of C. 6 

=10.61—10.61=0.00 


M. I), of C. 1 

“ “ 2 
“ “ 3 


M. D. Sta. 0+M. D. Sta. 1 

0+5.34 

— 2 

2 

M. T). Sta. 1+M. D. Sta. 2 

5.34+13.73 

2 

2 

M. 1). Sta. 2+M. D. Sta. 3 

13.73+31.62 

2 

2 


2.670. 

9.535. 

22.675. 


i 















































122 


M. D. of C. 4= 


A MANUAL OF LAND SURVEYING. 

M. D. Sta. 3+M. D. Sta. 4 _ 31.62+22,61 ^ 


2 


u u r _ M. D. Sta. 4+M. D. Sta. 5 22.61+10.61 

2 2 


3 


<i 


u g_ M. D. Sta. 5+M. I). Sta, 0 _ 10.61+ 0.00 _ g 


We may now put the whole matter in compact tabu¬ 


lar form as follows. 


Sta. 

Bearing. 

! Distance. 

1 

Corrected 

Latitude. 

Corrected 

Departures. 

1 

ci 

<73 

O 

q 

S 

M.D.of 

Course. 

03 

0> 

• 

• 

S. Area. 




N. 

S. 

E. 

W. 





0 

N. 26ir° E. 

12.00 

10.76 


5.34 



2.67 

28.7292 


1 

N. 59° E. 

9.80 

5.07 


8.39 


13.73 

9.535 

48.34245 


2 

S. 66° E. 

19.60 


7.94 

17.89 


31.62 

22.675 


180.0395 

3 

S. 35 0 W. 

15.68 


12.82 


9.01 

22.61 

27.115 


347.6143 

4 

S. 66 ° W. 

13.12 


5.32 


12.00 

10 61 

16.61 


88.3652 

5 

N. 46° W. 

14.72 

10.25 



10.61 

0.00 

5.305 

54.37625 



— — - — - 4 

26.08 26.08 31.62 31.62 131.4479 616 0190 


131.4479 

484!5711 
= Acres 48.45711 

In this example the area of the tract is evidently 
equal to the sum of the areas of the trapezoids c d and 
e based on courses 3, 4, and 5 minus the sum of the 
areas of the triangles and trapezoid a b and / based on 
courses 1, 2, and 6. 

The area of the triangle a equals the M. D. of course 
or line 1 multiplied by its latitude = 2.67 x 10.76. 

The area of the trapezoid b equals t he M. D. of course 
2 multiplied by its latitude — 9.535 X 5.07. 

in a similar manner we find the area of each triangle 

p 

and trapezoid. 

Examples for Solution : 

The following examples are taken from the field 
notes of the original United States Surveys in Michi¬ 
gan and are fair samples of the average work done on 
the government land surveys. The meanders of lakes 
and streams are run for the purpose of finding how 
much dry or uncovered land is contained in the ad¬ 
jacent tract to be paid for by the purchaser. 


N 

































COMPUTING AREAS. 


123 


Ex. 1. Meanders of a Lake in Section 5. 

Began at post corner to Sections 4, 5, 8, and 9, tlience 
in Section 5, N. 60° W. 6.50 ch. to S. E> Margin of Lake, 
thence in Sec. 5, N. 25° E. 4.00 cli., thence IN'. 51° W. 5.00 
ch., thence IN’. 18° W. 7.00 cli., thence N. 3° W. 7.00 cli., 
thence N. 63° W. 10.00 cli., thence S. 79° W. 6.00 ch., 
thence S. 7° W. 13.00 cli., thence S. 20° E. 6.00 cli., thence 
S. 6° W. 5.00 cli., thence N. 78° E. 14.00 ch., thence S. 
27° E. 5.00 cli., thence N. 71° E. 3.87 ch. to place of be¬ 
ginning on margin of Lake. 

Find the area of the lake. Also find the areas of the 
North and South halves respectively of the quarter sec¬ 
tion in which the lake lies, on the supposition that the 
quarter section is just 40 chains square and that the 
lines are run with the same variation of the needle as 
was used in meandering the lake. These areas are 
given in the official plat as follows: North 4, A. 66.18. 
South 4, A. 55.92. 

2. Find the area of the lake described in the exam¬ 
ple 13, page 91, also the area of each of the quarter- 
quarter sections adjoining the lake in the south half 
of Sections 11 and 12. These areas are marked in the 
official plat as follows : In Section 11, S. E. 4 of S. E. 4 
A. 31.50, N. E. 4 of S. E. 4 A. 20.40. In Section 12, S. W. 
4 of S. W. 4 A. 37.61, N. W. 4 of S. W. 4 A. 27.10. The 
meander post at the beginning of the survey is 14.00 
chains North from the Section Corner. 

3. Meander of a Lakq in section 2. 

Began at quarter post in line of Sections 2 and li, 
thence North 10.00 cli., to S. margin of Lake, thence in 
Sec. 2, thence S. 57° E. 13.00 ch., thence E. 3.00 ch., 
thence N. 45° E. 5.00 ch., thence N. 4° W. 6.00 ch., thence 
N. 70° W. 15.00 ch., thence S. 80° W. 6.00 ch., thence 
S. 244 E. 7.17 ch., to place of beginning in margin of 
Lake. 

Find the area of the Lake also the area of the W. 4 of 
S. E. 4 of Section 2 and of the S. E. 4 of the S. W. 4 ot 
the Sect ion. The first is given on the official plat as A. 
62.88 and the latter as A. 38.95 


m 


A MANUAL OF LAND SURVEYING. 


13. Problem. — Given the hearings of the sides of a 
field, to find the hearings when the field is supposed to be 
revolved so as to cause one of the sides to coincide with a 
meridian. 

EXAMPLES. 

1. The bearings of the sides of a field are, 1st, X. 12° E., 
2d, IS. 83i° E., 3d, 8. 21° W., and 4tli, ]S T . 47° W. What will 
the bearings be, if the field be supposed to be revolved so 
as to cause the first side to be on a meridian? 

Ans. —1st, X., 2d, N. 71i° E., 3d, 8. 9° W., and 4th, 

X. 59° W. 

Suggestion.—S uppose the field to be revolved toward the left, 
through an angle of 12°. Accordingly, each bearing would he changed 
by that amount. The readings of the new hearings are readily deter¬ 
mined by inspection. 

2. The bearings of the sides of a field are 1st S. 3p W., 

2d X. 86^° W., 3d 1ST. 16J° E., and 4th E. Required the 
new bearings when the first side is made to coincide with 
the meridian. 

Ans. —1st 8., 2d W., 3d X. 13° E., and 4th X. 86|° E. 

3. The bearings of the sides of a field are 1st 8. 20° W., \ 
2d S. 70° W., 3d X. 31° W., 4tli X. 45° E., and 5th 8. 60° E. 
Required the new bearings when the third side is made 
to coincide with the meridian. 

Aws.—lst 8. 51° w., 2d X. 79° W., 3d X., 4th X. 70° E., 
and 5th 8. 29° E. 

4. The bearings of the sides of a field are, 1st X. 45° E., 

2d 8. 30° W., 3d S. 5° E., 4th W., ahd-5th X. 20° E. What 
will the bearings become, if the field be revolved so as to 
•bring the third side to the meridian ? 

Ans. —1st X. 50° E., 2d 8. 35° W., 3d 8., 4th X. 85° W., 
5th X. 25° E. 

5. The bearings of the sides of a field are, 1st E., 2d 
X. 9° E., 3d 8. 69° E., 4th S. 66° E., 5th 8. 42° W., 6th 
8. 75° W., 7th X. 39° W., and 8th X. 42° E. What will the 
bearings become, if the field be revolved so as to cause 
the fourth side to coincide with the meridian ? 

Am.— 1st S. 24° E., 2d X. 75° E., 3d 8. 3° E., 4th 8., 5th 
X. 72° W., etc. 

Additional exercises may be formed from the above by requiring 
different sides to be brought to coincide with the meridian. 




COMPUTING ABE AS. 


125 


Rule. —Change each bearing agreeably with the direc¬ 
tion in which the field is supposed to be revolved by an 
amount equal to the bearing of the side which is brought 
to the meridian , and express the result in accordance with 
the proper form of denoting bearings. 

6. What were the bearings of the sides of a held which 
are now X. 16|° E., E., S. 31° W., and N. 86|° W., the vari¬ 
ation of the needle having changed 2|° toward the west 
since the former survey ? 


Supplying Omissions. — Erom inaccessibility of 
lines and sometimes from accident, omissions may occur 
in the held notes of a survey. In a closed survey, any 
two omissions may, in general, be supplied by computa¬ 
tion. It is, however, desirable to avoid as far as possible 
the necessity of supplying omissions in this manner, since 
it infringes upon the tests which otherwise serve to verify 
the work. 

The several cases which may occur are presented in the 
following problems: 


14. Prob. 1. To find an omitted bearing and distance. 

Case 1 . — When the omissions pertain to the same course. 

In a closed survey, the sum of the northings should 
equal the sum of the southings; and the sum of the east¬ 
ings should equal the sum of the westings. The defect 
of these equalities in the present case must be on the one 
hand the latitude and on the other the departure of the 
omitted course. 


Example.— 


Sta. 

Bearing. 

Dist. 

t* 

Dep. 

A 

N. 31° W. 

9.40 

+8.057 

— 4.841 

B 

N.45° E. 

9.30 

+6.570 

+ 6.576 

C 

Omit 

ted. 



E 

S. 20° W. 

5.30 

—4.980 

— 1.813 

F 

S. 70° W 

10.90 

—3.728 

—10.243 













126 


A MANUAL OF LAND SURVEYING. 


Solution.— Sum of 

northings = 14.633 
of southings = 8.708 

Diff. = CG = 5.925 S 
Sum of 

westings = 16.897 

of eastings = 6.576 

Dili = GE = 10.321 

The latitude of the omitted course is thus a southing 
and its departure, an easting. Its bearing is therefore 
S. — 0 E. 

To find the bearing or angle GCE, we have 

GE 10.321 

tan GCE = — =-= 1.74194. 

CG 5.925 

Whence, GCE = 60° 8'; or the required bearing is 
S. 60° 8' E. 

To find the distance CE, we have 

CE = (5.925 2 + 10.321 2 )* = 12.00. 

Remark.—I t will be noticed that a plat of the field may be made, i 
and the area found without supplying the omissions. 

Case 2. — When the omissions pertain to different 
courses. 

If the field be supposed to be revolved until the side 
whose length is omitted becomes a meridian, the given 
bearings being changed accordingly (Art. 13, Prob.), then, 
since the departure of the side made a meridian is 0, the 
difference between the sums of the eastings and westings 
of- the other courses is the departure, in its new position, 
of the side whose bearing is omitted. 



















COMPUTING AREAS. 127 

Knowing the length and the departure of this side, its 
latitude and bearing may be found, (Art. 8). 

The difference between the sums of the northings and 
southings of the courses in their new positions, is the 
length of the side which was made a meridian. 

Example .— 


Sta. 

Bearing. 

Changed 

Bearing. 

Distance. 

Lat. 

Dep. 

A 

N. 20° E. 

North. 

Omitted. 


0.0000 

B 

N. 45° E. 

N.25° E. 

8.00 

+7.2505 

+3.3809 

C 

S. 30° W. 

S. 10° W. 

5.00 

—4.9240 

—0.8682 

D 

Omitted. 


7.20 



E 

West. 

S. 70° W. 

5.92 

—2.0248 

—5.5630 


Solution .—Sum of eastings = 3.3809 
“ “ westings = 6.4312 


Difference = 3.0503 (an easting). 

Latitude of DE = (7.20 2 — 3.0503 2 )^ = 6.5219 (a southing). 
Sine of changed bearing of DE — 3.0503 - 5 - 7.20 = 0.42365. 
Whence “ “ “ DE is S. 25° V E. 

Whence original “ “ DE was S. 5° E. 

Sum of northings = 7.2505 
“ “ southings = 13.4707 


Difference = 6.22 = length of AB. 

Remark.—I t is sometimes doubtful whether the latitude of the 
course whose bearing is omitted is a northing or a southing. 

m the present case, the question is determined by a simple inspec¬ 
tion of the latitudes, since the sum of the southings is less than the 
sum of the northings, without considering the northing of the first ^ 
course. 

Tn other cases, there may be two sets of values of the omitted parts, 
with either of which the problem is satisfied. 

Practically, however, the ambiguity is removed by a general knowl¬ 
edge which the surveyor has of the directions of the lines. 
















128 A MANUAL OF LAND SURVEYING. 

15. Prob. 2. To find the omitted lengths of two courses. 

Case 1. — When the courses are consecutive. 

The bearing and length of a line which would close a 
survey, leaving out the unknown sides, may be found by 
Prob. 1, Case 1. This line and the unknown sides form a 
triangle in which the angles, as found from the given 
bearings, and the length of one side are known. The 
lengths of the other sides may therefore be computed. 

The procedure will be readily worked out by the stu¬ 
dent, without illustration. 

Case 2. — When the courses are not consecutive. 

This case may be treated in the same manner as the 
preceding. 

Or, we may suppose the field to be revolved so as to 
make one of the sides whose length is omitted, a merid¬ 
ian, the bearings of the other sides being changed accord¬ 
ingly. 

We may then find the difference of the sums of the 
eastings and westings, which will be the departure, in its 
new position, of the other side whose length is wanting. 

Having the bearing of that side and its departure, its 
length and latitude may be found. Finding the differ¬ 
ence between the sums of the northings and southings, 
we obtain the length of the side which was made a 
meridian. 

Example.— _ 


Sta. 

Bearing. 

Changed 

Bearings. 

Distance. 

Lut, 

Dep. 

A 

N.15° E. 

N. 30° W. 

5.00 

+ 4.33 

—2.50 ! 

B 

N. 45° E. 

North. 

Omitted. 


0.00 

C 

S. 55° E. 

N. 80° E. 

10.05 

1.75 

+9.90 

T) 

S. 15° tV. 

S. 30° E. 

12.25 

—10.61 

+6,12 

E 

S. 75° W. 

S. 30° W. 

Omitted. 



F 

N.33-M°W. 

N. 78%° W. 

9.96 

-f 1.95 

—9.77 

























COMPUTING AREAS. 


129 


Sum of eastings = 16.02 
“ “ westings = 12.27 


Difference = 3.75 = Dist. X sin 30°, 
Whence, length of EF = 3.75 -4- 0.5 = 7.50. 
Lat. EF = 3.75 -4- tan 30° = 6.50. 

Sum of northings = 8.03 
“ “ southings = 17.11 


Difference = 9.08 == length of EC. 

Remark.— If the sides whose lengths are omitted are parallel, the 
problem is indeterminate. 


16. Prob. 3. To find the omitted hearings of two 
courses. 

We find, (Prob. 1, Case 1) the bearing and length of a 
line which would close a survey, having the lines whose 
bearings are given as the other sides. 

The line thus found and the two lines whose bearings 
are omitted form a triangle. The lengths of the sides of 
the triangle being known, its angles may be found; and 
from the angles and the bearing of one of the sides the 
bearings of the other sides may be found. 

The closing line and the triangle are illustrated by the 
diagram accompanying the following 


Example .—I 


Sta. 

Bearing. 

Dist. 


Pep. 

A 

N.15° E. 

5.00 

-f 4.8296 

4-1.2941 

n 

Omitted. 

9.08 


# 

c 

S. 55° E. 

10.05 

— 5.7645 

4-8.2325 

D 

S. 15° W. 

12.25 

—11.8327 

—3.1705 

E 

Omitted. 

7.50 



F 

N. 33 3 4° W. 

9.90 

+ 8.2814 

—5.5334 


10 























180 


A MANUAL OF LAND SURVEYING. 


The side EF, without change of bearing, is represented 

by CG. BG is the 



Fig. 49. 


closing line of the 
field ABGHF , in 
which we have 


Sum of 

northings = 13.1110 
southings = 17.5972 


Difference = 4.4862 
(a northing). 


Sum of 

eastings == 9.5266 
westings = 8.7039 


Difference = 0.8227 
(a westing.) 


Whence (Prob. 1), bearing BG is X. 10° 23 / 30" W., and 
length BG is 4.56. 

In the triangle BGC, BC = 9.08 and CG = EF= 7.50. 


Solving the triangle, we find 

angle GBC = 55° 25' 40", ana angle BGC = 94° 3P 49". 

Whence, bearing BC is X. 45° V 10" E., and bearing 
EF is S. 75° 4' 41" W. 

Remark.—T he problem may possibly have two solutions, accord¬ 
ingly as the triangle may fall on either side of the closing line. The 
ambiguity is, however, practically unimportant. 


Exercises .—To be made by the student in the field. 


17. Most of the foregoing problems for finding areas 
may be simplified and much labor saved in calculation, 
by reducing the irregular polygons and oblique triangles 
to right triangles and trapezoids on the plat, and taking 
their dimensions by direct measurements from the plat, 
instead of calculating them. If the plat is made on a 
large enough scale—showing not more than four chains 





COMPUTING AREAS. 


131 


to the inch—and the drafting is carefully done, the meas¬ 
ures on the plat will be very nearly if not quite as good 
as those taken on the ground, and will give results suffi¬ 
ciently close for most purposes. 


1st Method.— Draw a diagonal between two distant 
angles of the figure, and perpendiculars to it from the 
other angles. 



C 



Fig. 51. 




























132 A MANUAL OF LAND SURVEYING. 

1. To reduce the trapezium abed (Fig. 51) to its equiva¬ 
lent triangle. 


Produce the line ab an indefinite distance. AVith the 
parallel ruler, or straight edge and triangle, find the point 
e, where a line through d parallel to ea intersects the line 
ab. Draw the line ec, intersecting ad at g. 

Then the triangle ecb is equivalent to the trapezium 
abed, for the triangles aed and ace, having the same base 
ac and equal altitudes, are equal; and the triangle aeg 
being taken from both leaves the triangle eag , which is 
added to the original figure, equal to the triangle edg, 1 
which is taken from it. 


The perpendicular may now be drawn from c, and the 
base eb and altitude fc measured on the plat. 

2. By an extension of the same process, any polygon 
may be reduced to one or more equivalent triangles. It 
will frequently be found convenient to divide the figure 
into two or more parts, and reduce the sides separately. 
The process is indicated in Figure 52. 


s 



Fig. 52. 


Let abedefgh be the polygon to be reduced. Extend 
one side, as ab, indefinitely for a base. From c draw ei 












COMPUTING AREAS. 


133 


parallel to bd. Prom d draw dk parallel to ei. From e 
draw el parallel to fk. Having selected f as the vertex 
of the triangle, we next draw fl for one of its sides. 

Next, from h draw hm parallel to ga. 

Frofn g draw gn parallel to fm. 

From/draw/n. for the third side of the triangle, and 
fo t its altitude. 

The tri angle fin is equivalent to the polygon abcdefgh. 
It is best to draw all these lines lightly on the plat, to 
avoid errors. 

If we consider each point, i, k, l, marked in succession 
on the base as an angle of the polygon, which it is until 
its successor is located, we have the following 

General Rule.— Extend one side indefinitely as a 
base. Commencing at the first angle from the base, draw 
from it to the base a line parallel to a line joining the two 
adjacent angles of the polygon. Continue drawing lines 
to the base from each angle in succession as far as re¬ 
quired. Join the last angle from which a parallel was 
taken , with the last point of intersection on the base, for 
a side of the final triangle. 

It is sometimes more convenient not to produce one of 
the lines of the figure for a base, but to draw a perpen¬ 
dicular to it from one end or from the end produced. 
The same rule applies. 

18. The preceding methods of taking measurements 
from the plat are found very convenient in estimating 
the area of land benefited by drainage, under the drain 
laws. Surveyors are frequently called on to make surveys 
and maps of drainage districts, showing the location of 
the drains and the location and area of the lands, belong¬ 
ing to the various owners, which will be benefited by the 
drainage. In most, if not all these cases, no man can tell, 
either before or after the drainage has been executed, just 
exactly where the dividing line is, between land which is 
benefited and that which is not benefited. For this rea¬ 
son a rapid survey of the approximate line, by stadia 


13 d A MANUAL OF LAND SURVEYING. 

measures, is just as good as tlie most elaborate work with 
the chain or tape. The one is likely to get as near the 
true dividing line as the other. 

The writer has found the following method to work 
well in his practice. Suppose a tract of marsh or swamp 
is to be measured and mapped, having more or less 
cleared upland around it: 

Assume some line as a base. A section line or quarter 
line of the United States Survey answers well for this 
purpose. From this base run a broken line around the 
swamp wherever it is most convenient to do so. Set a 
stake at each angle in the line. Note the length of each 
course and the angle which it makes with the common 
base, as described on page 83. 

When the circuit of the swamp has been made, and the 
transit again set up at the starting point, the work will 
prove itself. After taking a back sight on the last sta¬ 
tion and pointing the telescope along the base line, if the 
work has all been correctly done, the vernier should give 
the same reading as it did to start with, showing that 
just 360° have been passed around. 

In passing around the swamp an assistant with the 
stadia rod follows its margin, setting up his rod at every 
point where it changes its general direction. The transit- 
man notes down the direction of each point at which 
the rod is set up, by its angle from the base line and its 
distance from the transit as read oft - from the rod. 

When as many points are taken as are convenient from 
one station, the transit is moved up to the next one, and 
the operation continued till all the desirable points are 
located. This being done in the field, they are reproduced 
on the plat on a scale large enough to permit measure¬ 
ments on the plat with a reasonable degree of accuracy. 
The points along the margin of the swamp having been 
laid down on the plat, are connected by straight lines, and 
all intersecting farm lines or other points of interest are 
also laid down. 


COMPUTING AREAS. 


135 


We now have a map, showing as correctly as it is pos¬ 
sible to do so, the location of the swamp on each man’s 
land. The areas of the several tracts are found by taking 
the parallel rule and needle point and reducing these 
irregular polygons to their equivalent triangles and rect¬ 
angles, making the necessary measures on the plat and 
computing the areas from these measures. 

19. Division and Partition of Land. —The sur¬ 
veyor is sometimes called on to divide areas into portions 
having a specified relation to each other, or to part off 
from a field a given number of acres by a line fulfilling 
some specified condition with respect to the field divided. 

There is a great variety of these problems, most of 
which occur very rarely in the surveyor’s practice. A 
few of those which occur most frequently are given. 

Prob. 1. — To divide a triangle into parts having a 
given ratio. 

Case 1 .—By lines from an angle. 

Solution .—Let ABC be any triangle, and suppose it is 

required to divide it by a line from 
B, into two parts having the ratio 
of m to 7i. 

Let BD be the line of division, so 
c that ABB : BBC . : m : n (1) 

But ABB : BBC :: AB : BC (2) 

Combining (1) and (2), we have 

AB : BC :: in : n, 
whence, AB : AC :: in : m-\-n, 

m X AC n X AC 

whence, AB — -. Similarly, BC —-. 

m -f - n i n - f- n 

Measure the distance AB thus found, and run the line 
BB. 






136 


A MANUAL OF LAND SURVEYING. 


If the triangle were to be divided into three parts in 
the ratio of m : n : p, we should have 

m X AC n X AC 

AD — -and DE =-. 

m + n -\-p m + n -\~p 

Cor. —To part oft' by a line, as BD , a given area a, we 

a X AC 

have AD : AC :: a : area ABC , whence AD — -. 

area ABC 

Examples— 1. Find the measurements required to di¬ 
vide a trianglar field by lines from an angle to a side 
whose length is 12.30 ch., into parts to each other as 2, 3 
and 4. 

2. Find the measurement required to part off 3.5 acres 
from a triangular field a side of which is 18.50 ch., and 
a perpendicular thereupon from the opposite angle is 
10.40 ch. 

Case 2.— By lines parallel to a side. 

Solution. — Let D be the point in the side AB from 

which a line parallel to BC shall 

divide ABC so that ADE : DECB 

* 

:: m : n. Then 

ADE : ABC :: m : m -j- n. 

But ADE : ABC :: AD 2 : AB 2 , 

FlG - whence, AD 2 : AB 2 :: m : m -f n, 

i m ) * 

giving AD = AB j - C 

( m -f n ) 

Measure the distance AD thus found, and run DE 
parallel to BC. 

If the triangle is required to be divided into three parts 
in the ratio of m : n : p , we should have 
( m ) * 

AD — AB j-[ and AF — AB 

( m -f n + p ) 

Cor . 1.— To part off a triangle, as ADE, of given area a 


m A- 7i | ** 


{ m A- 7i -\-p 











GOMFUTING AREAS. 


137 


we have AD — AB 


area 


a ) * 
ABC ) * 


Cor. 2.—To part off a quadrilateral, as DECB, of given 
area, a', we may find by Cor. 1 the distance AD required 
to part off a triangle of the area ABC — a' and measure 
BD ~ BA — AD. 


Examples. —1. Find the measurement for dividing a 
triangular field of 12 A. into parts in the ratio of 4 to 5 
by a parallel run from a point in a side whose length is 
10.35 cli. 

2. Find measurements for dividing by parallels, the 
above field into three equivalent parts. 

3. Find measurement for parting off from the same 
field by a parallel, a triangle of 5 A.; a quadrilateral of 

7 % A. 

Case 3.— By lines perpendicular to a side. 

Solution.— Let ABC be a triangle required to be divided 

by a perpendicular to AC, into parts 
having the ratio of m to n. 

Let EF be the line of division, so 
that A EF : EBCF :: m : n, or 
AEF : ABC :: m : m-\-n. (1) 

Let BD be a perpendicular upon A C 
Then AEF: ABC :: AFX EF.ACX BD::m:m + n. (2) 
From similar triangles, AF : EF : : AD : BT ), 

afxbd 

whence, EF = -. 

AD 

Substituting this value of EF in (2), we have 



AF 2 X BD 

-: AC X BD :: m : m + n, 

AD or AF 1 : AC X AD v.m'.m + n 


whence, AF — 


AC X AD X w 
m -f - n 




Find AD and then AF. Measure the distance AF and 
run FE perpendicular to AC. 








138 A MANUAL OF LAND SURVEYING. 

Similarly, may be found the distances to perpendiculars 
dividing the triangle into three or more parts having a 
given ratio. 


Cor.—To part off a triangle, as AEF, of given area, a, 

(ACX AD X a)* 

we have AF = ] -£ . 

( area ABGD ) 

The distance AF to a perpendicular which shall part 
off a triangle AEF = a, may be found otherwise, as 
follows: triangle AEF = £ AF X FJF = and EF = 

( 2a )X 

AF X fan A. Whence, AF = \[ . 

f tan A ) 


Examples. —1. The bearings and lengths of two sides 
of a triangular field from the same corner are N. 20° E., 
15 ch., and N 50° E., 20 ch. Required the measurement 
from that corner to a perpendicular upon the longer side 
which shall divide the field into.two parts having the ra¬ 
tio of 2 to 3. 


2. Required the measurement to a perpendicular which 
shall divide the above field into two equivalent parts; 
into three equivalent parts. 


3. Required the measurement to a perpendicular which 
shall part off from the same field a triangle of 4 A.; a 
quadrilateral of 5 A. 

20. Prob. 2. To divide a trapezoid into parts having 
a given ratio. 

Case 1.— By lines dividing the bases proportionally. 


Solution .—Let ABCD be any trapezoid required to be 
divided into parts having the ratio of m : n : p. 

This is done in the easiest manner by dividing each 

base into parts having the ratio to 
each other as m, n and p, and join¬ 
ing the corresponding points of 
division. The measurements nec¬ 
essary to find the points of division 
are: 











m X BC 


COMPUTING AREAS. 


139 

m X AD 


BE = 


m -\-n + P 

n X AD 
and FH = -- 


EG = 


n X DC 


m + n -\-p 


AF = 


m -f w -f .P 


m + n + P 


Cor. —To part off a given area a by a line, as FF, 
which shall divide the bases proportionately, we have 

- a X DC a X AD 

BE =-and AF — -. 

area ABCD area ABCD 


Examples.—!. Given AD, N. 80° E., 12.60 ch., AB, 
' N. 10i° E., 8.12 ch., and BC, X. 80° E., 10.34 ch., to find the 
measurements required in dividing the field into parts 
having the ratio of 4 to 7, by a line dividing the parallel 
sides proportionally. 

2. Find the measurements for parting off from the 
above field an area of 5 A., by a line dividing the parallel 
sides proportionally. 

Case 2. — By lines parallel to the bases. 

Solution. —Let ABCD be a trape¬ 
zoid to be divided into parts in the 
ratio of m to n, by a line parallel to 
BC. 

Suppose EF to be the required 
line of division, so that 

EBCF : AEFD :: m : n. 

ltegarding the sides AB and DC as prolonged to meet 
at O, we have OAD : OBC :: AD 2 : JtC 2 , 
whence, OAD — OBC, 

or ABCD : OBC :: AD 2 — BC 2 : BC 2 . (1) 

Similarly, we have EBCF: OBC:: EF 2 —BC 2 :BC 2 . (2) 

Combining (1) and (2), ABCD: EBCF:: AD 2 — BC 2 : 
EF 2 — BC 2 , 

or m + n : m :: 



AD 2 — BC 2 : EF 2 — BC 2 . 












140 


A MANUAL OF LAND SURVEYING. 


fmX AJJ 1 -\-n X BC 2 \ X 

whence EF = «---s («) 

( m + n ) 

Supposing BH to be parallel to CD, the triangles ABH 
and EBG give AB : AH :: EB : EG, 

or AB : AD — BC :: EB : EF — BC. 

AB {EF — BC) 

Whence, EB =-. (6). 

AD — BC 

Thus, first finding EB by formula (a), we can then find 
EB by formula (b), and measuring that distance from B, 
we may run EF parallel to BC, dividing the trapezoid as 
required. 

Similarly, a trapezoid may be divided in three or more 
parts having a given ratio. Indeed, the above formulas 
may be directly applied to that purpose by making a 
simple substitution. 

Cor.—To part off a trapezoid of given area a, adjoining 
BC, we obtain from formula (a) 

(a X AD 2 + (area ABCD — a) BC 2 ) X 

EE = ^--C 

l area ABCD ) 

The distance BE is then found from formula (6). 

Examples. —1. Given a trapezoidal field ABCD in which 
AB is an east and west line, 9 ch., BC a north and 
south line, 5.19 ch., and AD a north and south line, 8 ch., 
it is required to run a north and south line dividing the 
field so that the parts on BC and AD shall have the ratio 
of 2 to 3 . 

2. Find the measurement from A to part off from the 
above field by a north and south line an area of 3 A. ad¬ 
joining AD. 

Case 3 .—By lines perpendicular to the bases 





COMPUTING AREAS. 


141 


Solution .—Let ABCD be a trapezoid to be divided into 

parts in the ratio of in to n by a line 
perpendicular to AD. 

Let EF be the line joining the 
middle points of the non-parallel 
sides AB and CD. We divide EF, 
as at G, into two parts having the 
ratio of in to n, and through G run If I perpendicular to 
AD. 

To find the point G on the ground, we have the form- 
m(BC -f- AD) 

ula EG =-—-. Whence, measuring from E the 

2 (in + n ) 

distance EG on the bearing of BC, we have the point 
sought. 

Cor.—To part off a given area a, by a line perpendic- 

a {BC + A D) 

ular to the bases, we have EG — ---, 

2 X ar ea ABCD 

Or, denoting the altitude of the trapezoid by h, we 
a a 

have EG — — =-——-. 

h AB X sin A 

The point I or If may be found by the formula 
AI=EG -f AE X cos A, or BH — EG — EB X cos A. 

Examples. —1. Given AD, E. 20 ch., AB, N. 15° E., 
9.50 ch., and BC y E. 12 ch., required the measurement for 
dividing the field by a perpendicular to AD into two parts 
having the ratio of in to n. 

2. Required the measurement for parting off from the 
above field, by a perpendicular to AD, an area of 4 A. 
adjoining AB. 

21. Prob. 3. — To divide a trapezium into parts 
having a■ given ratio. 

Cask 1.— By lines from an angle. 









142 A MANUAL OP LAND SURVEYING. 

Solution— Let ABCD be a trapezium to be divided into 

two parts having the ratio of m 
to n, by a line from C. 

We draw AC, and from B draw 
a line parallel to AC, meeting DA 
produced at E. We then divide 
ED, as at F, into the parts EF 
and ED, having the ratio of m to 
n. The line CF divides the trapezium as required. That 
is, ABCF : FCD :: m : n, or ABCF : ABCD :: m : m -f- n. 

Sch.—T he above solution is readily executed on the ground. 



Fig. 59. 


In a similar manner a trapezium may be divided into 
any number of parts having a ^.iven ratio. 

The point F may be otherwise found as follows: 

n X ABCD 

The triangle DCF — -, 

m -f n 

DC X sin D X VF 
and again, DCF — -. 

9 

Li 

2n X ABCD 

Whence, DF= -. 

DC (m + n) sin D 

Cor.—To part off a triangle, as DCF , of given area a, 

2 a 

we have DE —-. 

DC X sin D 

Examples— 1. Given AB, N. 8° W., 7.60 ch., BC 
N. 76i° E., 10.21 ch., CD, S., 11.40 ch., and DA, N. 80*° W. 
9.00 ch. Required the measurement for locating a line 
CF which shall divide ABCD into the parts ABCF and 
FCD, to each other, respectively, as 2 to 3. 

2. Required the measurement for parting off from the 
above held a triangle DCF of 10 A. 







COMPUTING AREAS. 


143 


Case 2. — By lines parallel to a side. 


Solution—"Let it be required to divide a trapezium, as 

A BCD, by a line, as EF, parallel 
to AD, into two parts, EBCF and 
AEFD, to each other as m to n. 





Suppose the sides including the 
parallel to be produced to meet 
at 0. The triangle BOC may be 
regarded as known. Call its area 
a. The trapezium EBCF is known as to area, being 
m X ABCD 

-. Call this area b. 

m -f n 

The area of the triangle AO/) is known. Call it e. Its 
side AO is also known. 

((«t 6)^ 

Now, (Art. 19, Prob. 1, Case 2), OE = AO j- 

Whence, BE = OE — OB. 

Measure this distance and run EF parallel to A D. 


Another procedure is to draw BI parallel to AD, form¬ 
ing the triangle BCI, whose area and side BI may be 
found; whence the ratio of the trapezoid EBIF to the 
trapezoid ABID is obtainable, and accordingly the dis¬ 
tance BE. 

Sen.—The problem of parting off a given area from a trapezium by 
a line parallel to a side, is essentially the same as the above. 


Examples— 1. The field being as given in Ex. 1, Case 1, 
it is required to find the measurement for locating a par¬ 
allel to BC that shall divide the field into two parts in 
the ratio of 3 to 4. 

2. Find the measurement required to part off from the 
same field an area of 10 A., by a line parallel to BC. 






144 A MANUAL OF LAND SURVEYING. 



22. Prob. 4. Two men own land situated between a 

road XX' and a 
line YY', and di¬ 
vided by a line 
BA'. 


It is required to 
run a line AB', at 
right angles with 
the road, which 
shall part off areas 
of equal value from 
the two portions. 


Fig. 6i. 


Solution. —Let T be the % triangle AOB, and T' the tri¬ 
angle A'OB'. 

Let v = value per acre of T, and v' revalue per acre 
of T'. 

Let angle OB A = B, and angle OA'B' = A' be known; 
and let AB = x, BA' — c, and BO = z. 

We shall then have 

z 2 sin B cos B 

area .7'=-, (1) 

2 

(c — z) 2 sin A' cos B 

■ and area T' =-, (2) 

2 cos (A' — B) 

By conditions of the problem, Tv = T'v'. 

Whence, T : T' ;; v' ; v. Let the ratio v' : v = r. 

Then T = T'r. Whence, from (1) and (2), 
z 2 r sin A' 


(c — z) 2 


sin B cos (A' — B) 



r sin A' 


\ H 


sin B cos (A' — B) 


= //. 


cn 

, and x — z cos A 


cn cos B 


Whence, z 


n 1 


li d - 1 













COMPUTING AREAS. 


145 


23. Many problems which the surveyor meets with 
may be readily solved by trial lines and successive 
approximations. A line is run or assumed to meet the 
required conditions as nearly as can be judged. The 
area parted off by the line is computed and the amount 
of error found. A new line is assumed to correct the 
error, and thus successive approximations to the true line 
are made until the error disappears. If good judgment 
is used, it is sometimes the quickest and easiest method 
to solve the problem. . 


Example .—The northwest quarter of Section 30 is di¬ 
vided by an angling road. The owner wishes it laid off 
into live acre lots, commencing at the south end, the lot 
lines to be parallel with the quarter line, and running 
from the center of the road west to the section line. Re¬ 
quired the number of lots, the area of the fractional lot, 
if any, at the north end, and the dimensions of the several 
lots. The total dimensions are given on the figure. 



small by .1 acre, which must 
li 


Solution, (First Lot).— 
Length of south line, 7.65 
ch. If the lot were a rect¬ 
angle of 9.00 chains base, 
the perpendicular ae would 
50.0000 

be -= 5.5554- chains. 

9.00 

Assume that ac — 5.60, to 
find cd. The line bx di¬ 
verges from ay at the rate 
23.38 — 7.65 

of-or .39325 ch. 

40.00 

per chain. Then cd = 
7.65 -f (560 X .39325) =9.852 
chains. Area abed — 

9.852 + 7.65 

_X 5.60 = 

2 

4.900056 acres. This is too 
>e added. 












146 A MANUAL OF LAND SURVEYING. 

For the next approximation we observe that the addi¬ 
tion of 1 link along the line cd adds nearly .01 A. to the 
area. So we will add 10 links for the trial. 5.60 -f- .10 = 
5.70, and 5.70 X .39325 = 2.2415 = divergence of lines. 
2.2415 + 7.65 + 7.65 

-— X 5.70 = 4.99933 A. The result is 

2 

still a trifle short, but in ordinary surveying would be 
sufficiently correct. 

To find the remaining side of the lot, bd, we have a 
right triangle, with a base equal to ac and perpendicular 
equal to cd — ah. 

The method is now sufficiently described so that the 
student may finish the computations and make a plat of 
the example. 

Field Notes.—Nearly every surveyor has a method 
of his own for keeping the field notes of his surveys. 
For general purposes probably no better plan has been 
devised than that employed in the United States land 
surveys. This method gives, in a condensed narrative 
form, each item in the survey, in the order in which it 
was executed, and affords opportunity for explaining all 
the details as fully as may be .necessary. 

It is a common fault among surveyors to condense 
their notes into the least possible space by omitting many 
things of importance and by the use of arbitrary signs, 
which may or may not be understood by any one else who 
may have need to refer to them. The notes are thus de¬ 
prived of much of their value, and in case it were desired 
to use them as evidence in the courts, they might be 
excluded altogether. 

The field notes should be full and explicit, and, espec¬ 
ially in re-surveys, should state in plain, concise words 
every material fact in regard to the work done. Starting 
points should be described and identified; the direction 
of lines, how determined, whether from the true merid¬ 
ian, the magnetic meridian, or from an arbitrary meridian 



COMPUTING AREAS. 


147 


adopted for the line, should be shown. It is not enough 
to say that the survey started from a certain corner. 
That may be disputed, and the notes should give the 
evidence by which it is known to be the corner. Tell 
what was found to mark the corner. If a bearing tree of 
a former survey is found, give its direction and distance 
from the corner. Make everything so clear and plain 
that the average citizen can understand it and judge of 
the trustworthiness of the survey. The following is a 
sample of the field notes of the United States survey. It 
is an extract from the 


FIELD NOTES 

OK Til K SUli V K V (»K TH K 

SUBDIVISION AND MEANDER LINES 

OK 

Township No. fi North, Range No. 34 East 

OF THK 

PRINCIPAL BASE AND MERIDIAN 
OK 

MONTANA TERRITORY, 

as surveyed by 

Walter W. de Lacv, 

U. S. Deputy Surveyor, 

Under his Contract, 

No. 87, 

Dated July 3, 1880 


148 


A MANUAL OF LAND SURVEYING. 
T. 6 N., R. 34 E. 


ChainF. 


20.00 

31.00 

40.00 


52.70 

53.82 


68.82 


72 50 
80 00 


Preliminary to commencing this survey, I ran west on a blank 
line on the south boundary of Sec. 36, and at 39.97 chs. found the 

34 sec. cor. and at 80.01 chs. found the sec. cor. As the east 
boundary of Sec. 31 crosses the Yellowstone River it was not re¬ 
run. My compass will therefore run the same line as the exte¬ 
rior boundaries, and the chaining practically agrees. _ 

Survey commenced August Gtli, 1879, with a Burt’s improved 
solar compass. 

I commenced at the cor. to Secs. 1,2, 35, and 36, on the south 
boundary, which is a sandstone 30x8x2 1 4 ins. firmly set in the 
ground, with one notch on E. and 5 notches on W. edges, and 
pits 18x18x12 ins. in each sec. 534 ft. dist. with mound of earth 
2 ft. high, 434 ft. base alongside. Thence I run North bet. Secs. 

35 and 36. ” Va. 18°30'E. 

Enter scattering timber. Alexander’s house bears N. 31° \\. 
Leave scattering timber. 

Set a post 3 ft. long, 3 ins. square, with marked stone, 12 ins. in 
the ground, for 34 sec. cor., marked 34 S. on W. side, dug 
pits 18x18x12 ins. N. and S. of post 534 ft. dist., and raised a 
mound of earth 134 ft. high, 334 ft. base, around post. 
Alexander’s house bears S. 5324° W. 

Enter brush. 

Right bank of the Yellowstone River. Set a post 4 ft. long, 4 
ins. square, with marked stone, 12 ins. in the ground, for 
meander cor. to fractional secs. 35 and 36, marked M. C., ami 

T. 6 N. on S., 

R. 34 E. S. 36 on E., and 

S. 35 on W faces, dug pit 3 ft. square, 12 ins. deep, 8 Iks. 

S. of post, and raised mound of earth 2 ft. high, 434 
ft. base, around post. 

There being an island on line on N. side of channel, I send a 
flag across, and set it on line bet. secs. 35 and 36, on bar S. 
of island. I then go across to flag and run a base line W. 
11.14 chs., to a point from which meander cor. on right bank 
hears S. 37° 50' E., which gives for distance across the river 
to edge of bar 14.34 chs. I then run north from flag 66 Iks. 
to south bank of island, making the whole distance 53.82 + 
14.34 + 0.66 chs., or 

To south bank of island, which point I established by setting a 
post 4 ft. long, 4 ins. square, with marked stone, 12 ins. in 
the ground, for meander cor. to fractional secs. 35 and 36 on 
S. bank of island, marked M. C., and 

T. 6 N. on N., 

R. 34 E. S. 36 on E., and 

S. 35 on W. faces, dug pit 3 ft. square, 12 ins. deep, 8 Iks. 

N. of post, and raised a mound of earth 2 ft. high, 
434 ft. base, around post. 

Thence continue on line across island, enter brush. 

Leave brush, enter timber. 

Set a post 4 ft. long, 4 ins. square, with marked stone, 12 ins. in 
the ground, for cor. to secs. 25,26,35, and 36, marked 

T. 6 N. S. 25 on N. E., 

R. 34 E. S. 36 on S. E., 

S. 35 on S. W., and 

S. 26 on N. W. faces, with 1 notch on S. and E. edges, 
from which 

A cottonwood, 12 ins. diam., bears N. 1224° E., 180 Iks. 
dist., marked T. 6 N., R. 34 E., S. 25 B. T. 

A cottonwood, 18 ins. diam., bears S. 82° E., 154 Iks. 
dist., marked T. 6 N , R. 34 E., S. 36 B. T. 

A cottonwood, 10 ins. diam., bears S. 293i° W., 56 Iks. 
dist., marked T. 6 N., R. 34 E., S. 35 B. T. 

A cottonwood, 10 ins. diam., bears N. 4634° W., 119 Iks. 
dist., marked T. 6 N., R. 34 E., S. 26 B. T. 

Land, level. 

Soil, rich loam—1st rate. 

Timber, cottonwood and willow, undergrowth same, 12.30 ch. 







COMPUTING AREAS. 149 

The following is a sample of Field Notes of a 
Resurvey, kept upon the same plan: 


Survey on Section 14, Township 2 South, Range 10 West, 

For J. R. Comings and H. Rowland. 

May 22,1874. 

C. Rowland^’ J’ Chainmcn. S. Comings, Flagman. 

Commenced at the S. E. corner of Section 14. Found a piece of 
strap railroad iron driven for the corner, which Hugh Shatter says he 
knows to have been kept in the same place, unquestioned, as the corner 
for over 30 years. Marked 

a maple, 8 in. diam., S. 45° W., 77 Iks. dist. 
a burr oak, 12 “ “ N. 43° W., 123 “ 


Chains. 

40.00 

80.24 


I set up a tall flag on the corner and then ran west on random 
Va. 2° 15' E., setting temporary stakes every 10 chains in line. 
Quarter section corner lost. 

Intersected the west line of Sec. 14, 42 links south of the corner. 
Found rotten stake at correct point, N. 26° El, 104 Iks. from 
stump of wh. oak 24 in. diam., bearing tree of U. S. Survey, 
having surveyor’s mark distinct on it. Set a piece of steel 
T rail 28 inches long for corner. Marked 

locust, 16 in. diam., S. 28° W., 116 Iks. distant, 
burr oak, 18 “ “ N. 78° E., 152 “ 


Ran thence east on corrected line at single sight with transit, 
from corner to corner. Va. 2° 33' E. 10:30 A. M. 

40.12 Found cedar stake 3 feet below surface of road crossing and 2!4 
links south of line. No other evidence of corner to be found. 
Put a piece of T rail 24 inches long on top of the stake for 
quar. sec. cor., 55 links south of south rail of M. C. R. R. 
No tree near. 

60.18 Planted granite boulder 20x12x6 inches, with cross -f mark, for 
y 2 quarter corner, in true line between qr. post and section 
corner and marked 

maple, 12 in. diam., S. 16° E., 55 Iks. distant, 
burr oak, 16 “ “ N. 54°E., 118 “ “ 


In some surveys, such as laying out additions to cities 
or villages, or any similar work, it is better to make a 
rough sketch or plat of the work in the field book and 
mark the dimensions and directions of lines on the plat. 
Field books which are ruled in small cross sections are 
best adapted to this use. 

Abbreviations.— Where the work of the land sur¬ 
veyor consists in re-surveys and sub-dividing sections 
of the United States Surveys, the field notes may be 
made more concise and liability of error reduced by al¬ 
ways using a definite symbol to refer to each corner of 
the section or sub-division. The symbols should be 
simple and adopted upon some system by which 








150 


A MANUAL OF LAND SURVEYING. 


they may be easily remembered and located in the 
mind. 

The system shown in the figure has been used 
many years by surveyors in Michigan and found sat¬ 
isfactory. 

All the corners lying in the 
exterior lines of the section are 
numbered in a definite order of 
rotation in accordance with 
their relative importance. Let¬ 
ters are used for the interior 
corners, the first letters being 
used for the corners lying in the 
quarter lines and the others for 
the centers of the quarter sections. 

The following is a sample of the manner of using the 
symbols in keeping notes upon the U. S. System when 
sub-dividing a section. 


5 10 


10 

8 

15 



e 

a 

f 


d r 


b 


V 

h 

f 

c 

g. 






3 

11 

6 

12 


14 7 13 


80.22 


Began at 7. Found stake in place and both bearing trees stand¬ 
ing. Planted stone 25" X 8" X 0" marked -|- for corner. Thence 
north on random. Yar. 2° 30' E. setting temporary stakes 
every 10 chains 

Intersected Section line 20 links west of 5. 

At5 found rotten stake at correct point, S. 28° W. 00 Iks from stump 
of W. Oak bearing tree of U. S. Survey. 

Drove stake for corner and put broken earthenware and glass 
around it and marked 

Wh Oak 12" d ; N. 06° E. 42 Iks. 

Wh. Oak 18 N 34 W 63 Iks. 


39.92 

9.98 

19.90 


29 94 


From 5 ran east on random, setting temporary stakes every 10 
chains. 

Intersected Sec. line 12 Iks. North of 2. Found earthen post in 
correct position and bearing trees of resurvey standing. 
Thence West on corrected line. 

Set stake on true line. 

At 11 set stake with stones around it and marked 
Pine 12 N. 40° W. 79 Iks. dist. 

Bed Oak 24 S. 191° W. 72 dist. 

Set stake on true line. 


From 11 ran south on random Yar. 2° 19' E. and set temporary 
stakes at 20 and 40 chains. 


20.02 

40.18 

80.04 


39.99 


Then went to 0. Found post and bearing trees of resurvey stand¬ 
ing. Ran thence West on random Var. 2° 20' E. 

Intersected random from North 0 links South of temp, stake. 
Intersected random ^4 line 8 links North of temp, stake. 

Int. Sec. line 10 links South of 8. Corner post dug out in road. 
Set iron plow beam for corner S. 29 W. 76 Iks. from hearing 
tree of U. S. Survey. 

Thence East Corrected line. 

At intersection of quarter lines set post. 













CURVELINEAR SURVEYING. 


151 


CHAPTER TIL 

CURYELINEAR SURVEYING. 

1. As land surveyors have occasion in laying out 
streets in villages, parks, cemeteries, race courses, drains, 
etc,, sometimes to make use of curved lines, it has been 
deemed proper to include in this work a short discussion 
of the manner of locating the simpler curves, and add 
such tables as are needed for this use. For a more com¬ 
plete exposition of the subject, consult the field books of 
Henck, Trautwine, Shunk, or Searles. 

The curve most commonly used is the circular curve, 
simple or compound. 

The simple circular curve, as its name indicates, is a 
circle or an arc. When an arc is used to connect two 
straight lines, these lines, from their relation to the circle, 
are termed tangents. 

The compound circular curve is a combination of arcs 
having different radii. At the point of junction of any 
two of these arcs their radii lie in the same straight line. 

Of the several geometrical propositions on which the 
theory of running curved lines depends, it will not be 
necessary for our purpose to recall more than the fol¬ 
lowing 


PRELIMINARY PROPOSITIONS. 

A If a circle be drawn touching each of two intersecting 
lines at but a single point , then the exterior angle made 
by the intersection of these lines is equal to the angle at 
the center of the circle which is measured by the arc 
intercepted by the two lines at their points of tangency. 

2. The angle which either line makes with the chord of 
the intercepted arc equals one-half the angle at the centre 
of the circle which is subtended by that chord. 


152 


A MANUAL OF LAND SURVEYING. 

In Fig. 63 CF and 
TI represent the two 
lines tangent to the 
circle at C and T, and 
intersecting at I. The 
angle FIT — COT , 
and the angle FCT— 
V* COT. 

The angle FIT is 
called the deflection 
angle, and the angle 
FCT the tangential 
angle. 

Fig. 63. 

Curves are named from the angle which is subtended 
by a chord 100 feet long. Thus, if the 100 foot chord 
subtends an angle of 1 degree, the curve is spoken of as 
a 1° curve; if of 5°, as a 5° curve, and so on. Tables have 
been prepared giving the various functions of a 1° curve, 
which are of great assistance in running curved lines, 
saving nearly all the trouble of calculation. The foot is 
taken as the primary unit of these tables and is most 
commonly used, but any other unit using the decimal 
notation, as a link or metre, is just as readily applied. 

Curves are run on the ground by successive deflections 
of chords. The amount of each deflection may be meas¬ 
ured on the ground with the tape or turned off on the 
transit. 



2. To run a Curve with Pickets and Tape. 

—First, determine the radius of the curve and the length 
of chord to be used. The latter is usually 100. From 
these data the amount of deflection for each chord is 
determined as follows: 

chord 2 

I >ell. dist. = 


radius 


Tangential dist. = defl. dist 








CURVELINEAR SURVEYING. 


153 



Example 1. —Let ah be the straight line or tangent 
which is to be continued from b by a curve having a 
radius of 1,433 feet, using chords of 100 feet. 

Extend the line ab to c, making bc = fbd l — cd 2 . Ex¬ 
tend the chord bd to e, making de = bd — df. Extend 
the chord df in a similar manner, cbd is the tangential 
angle, and cd the amount of the deflection to be meas¬ 
ured from the tangent to find the line of the curve, edf 
is the deflection angle, and ef is the amount of deflection 
to be measured off from the extension of the chord bd to 
find the line of the curve. 

To find the distance ef. —The triangles edf and dof 

df 1 IOC 2 

being similar, ef : df :: df do. :. ef= — =- 

do 1433 

= 6.98 nearly. The tangential deflection being one-haif 
the chord deflection, cd = %ef — 3.49. The triangle bed 

is right-angled at c, hence be— f bd 1 — cd 1 — j/100 2 —3.49 2 

99.94. The difference between be and bd is so small 
that in all curves of large radius it may be neglected on 
the ground and be be measured off = bd. 

These lines may be run with pickets, the chords meas¬ 
ured with the tape, and the deflections when not too large 
measured off by a graduated rod or a rod cut to the exact 
length. 







154 A MANUAL OF LAND SURVEYING. 

Example 2 —Lay off on the ground a curve having a 
radius of 2,640 feet, using chords of 50 feet. 

Ex. 3 ,—Lay off a curve having a radius of 819 feet and 
chord of 50 feet. 

Ex. 4.—Lay off a curve*with radius 2,865 feet, chord 100 
feet. 

Ex. 5 .—Lay off a curve with radius 1,910, chord 100. 

Ex. 6 .—Lay off a curve with radius 882, chord 50. 

Ex. 7 .—Lay off a curve with radius 1,042, chord 100. 

3 . Keeping the Field Notes of Transit 
Lines.— The field notes of transit work where long lines 
are being run, as for railroads, drains, etc., are usually 
kept in a different manner from those of other surveys. 
The notes proper are kept on the left-hand page of the 
field book. The opposite page is used for explanatory 
matter, sketches of topography along the line, such as 
road and stream crossings and obstacles in line, in greater 
or less minuteness of detail according to circumstances. 
The line is marked by stakes driven at regular intervals, 
usually 100 feet or 100 links, and numbered from 0 up¬ 
wards. The corresponding numbers are kept on the left- 
hand column of the note book, commencing at the bottom 
of the page and running upwards. 

If the topography is sketched on the right-hand page, 
the number of every stake is put down in its regular 
order, and the ruling of the book forms a scale by which 
the sketches are made. A book ruled in cross-sections is 
very convenient for this work. If the topography is not 
taken, the important stations are noted down and the 
intermediates are omitted. The following abbreviations 
are used: P. I., point of intersection; P. C., point of curve, 
or point where the curve begins; P. C. C., point of com¬ 
pound curve; P. It. C., point of reverse curve; P. T., point 
tangent, or point where the curve ends; T. P., turning 
point, indicating wiiere the transit was set up, also indi¬ 
cated by O or A* 

The direction of the tangents is kept as shown by the 
magnetic needle. This serves as a check on the angles of 
deflection, and assists in locating errors. 


CURVE LIN E A R SCR VEY INC. 


155 


SPECIMEN OE ABRIDGED NOTES. 


[left page.] 

Notes of Line “B,” D. & R. G. 
mile above the Dead Horse 
satch Co., Utah. 


[right page ] 

W. R. R., commencing about a 
Crossing of Price River, Wah- 


Sta. 




30 o 

20 o 

P. C. 4° C 

P. T. S. 

urve r’t. 

80° E. 

Def 32° 20' 
15° 00' 

7° 30' 

19 



6° 00' 

18 



4° 30' 

17 



3° 00' 

16 



1° 30' 

15 o 

P. C. 3° C 

urve l'ft 


10 o 




4 




3 




2 




1 




0o 

8. 65° E. 







-)-60 Old Spanish! 


Trail. 


S. 70° W. 



at 17 -f~ ol .4. 



Indian trail /. 

Jr> 



20 ft. wide, 
10 ft. deep. 




4. To Run a Curve with the Transit.— The 

transit is set up on the point in the tangent from which 
the curve is to commence. The limb is clamped with the 
verniers at zero, the telescope ranged along the line of 
the tangent, and the instrument clamped in that position. 
The tangential angle, = % the deflection angle, is then 
turned off on the limb. The leading chain-man draws out 
the chain or tape in the desired direction, and when out 
at full length, places his rod in line as directed by the 
signals of the transit-man. He then carefully measures 
the length of the chord, marking the distance with his 
rod, which is then aligned the second time. A stake is 
driven to mark the point, and the chain-men go ahead 
and measure the second chord, being aligned by the 
transit-man as before, and thus continue as far as neces¬ 
sary or convenient. The transit-man turns off equal 


























156 


A MANUAL OF LAND SURVEYING. 


angles on the transit for each successive chord as it is 
measured. At the end of the last chord which is run 
from any one setting of the transit, a short stake is 
driven firmly into the ground and a tqck driven in the 
top of the stake, to mark the exact point. If the curve 
is to be continued, the transit is moved up to this point, 
and with the limb clamped as it was used at the last 
observation, the telescope is ranged back to the point 
from which the observation was taken, and the instru¬ 
ment clamped in that position. As the angles have all 
been turned off from a point in the circumference of the 
circle, they are only half as great as the angle at the 
center subtended by the same chords. Hence the transit- 
man now unclamps the limb and turns off as much more 
angle as he had previously laid off. This gives him a new 
line, tangent to the curve, from which he may continue 
to lay off chords as before. 

Some transit men, instead of doubling the angle after 
the back-sight is taken, turn off an equal amount in the 
opposite direction on the limb before taking the back¬ 
sight. Then, after getting the back-sight, the verniers are 
brought to zero on the limb, when the telescope will give 
the line of the new tangent, as before. 

Curves are usually run to connect two straight lines 
which have been previously located. In such a case, pre¬ 
liminary to running the curve, it is necessary to find— 

1st. The deflection angle between the lines. 

2nd. The radius of the curve to be used. 

3d. The P. C. and P. T. 

4th. The length of the curve. 

The manner of procedure in such a case is indicated in 
the following: 

Example 1— To join two straight lines having a deflec¬ 
tion angle of 48° 16', by a curve the middle point (/) of 
which shall be at a distance of 112 feet from the point of 
intersection. 

Assume that the line aba has been marked with stakes 
100 feet apart, and that the point of intersection is found 
to be at stake No. 116, -f 43.7; in other words, that the 
P. I. is 43.7 feet beyond stake No. 116. 


CURVELINEAR SURVEYING. 


157 



The transit is set up over the point of intersection, the 
verniers clamped at zero, the telescope reversed and 
ranged along the line ab, and the instrument clamped in 
that position. The telescope is then righted, the upper 
clamp loosened, the telescope turned and the limb again 
clamped with the telescope pointing along the line cde , 
and the angle read = 48° 16'. Before proceeding further, 
it is necessary to determine the degree of curve to be 
used. By the conditions of the example, the middle 
point of the curve is to be 112 feet from the 1\ 1. Turn¬ 
ing to the table of functions of a 1° curve, we find its 
external secant, cf, to be 548.8 feet for an angle of 48° 16'. 

548.8 

Dividing this by 112, we find-= 4.9, or 4° 54', to be 

112 

the degree of curvature to be'used. Next we find the 
distance be = cd, which is to be measured along the lines 
to find the beginning and end of the curve, the P. 0. and 
P. T. Referring again to our table, we find that the 
tangent of a 1° degree curve for a deflection of 48° 16' 
is 2567.1, which divided by 4.9, the degree of curvature, 
gives 523.9. 

We now measure from the P. I. 523.9 feet along the line 
cde, and set a hub and drive a tack in it for the P. T. In a 
similar manner we next locate the beginning of the curve, 
which, subtracting 5 -f- 23.9 from 116 -f* 43.7, we find to be 
at Station 111, + 19.8. If the ground be clear and open, 






158 A MANUAL OF LAND SURVEYING. 

so that the whole curve may be seen at once, the transit 
may now be set up on the P. T., and the whole curve and 
as much of the next tangent de as desired run at one 
setting of the instrument, at the same time avoiding 
most of the errors usually made in running the curve 
from the P. C. If this cannot be done, the transit is set 
up at the P. C. with verniers at zero and a foresight on 
the P. I., or back-sight to some point along the line ab. 
The P. C. being at Sta. Ill, + 19.8, the first deflection will 
be for the partial chord found by subtracting 19.8 from 
100 = 80.2, or .802 of the full deflection. The tangential 
deflection for a full chord being 2° 27% for the partial 
chord would be .802 of 2° 27'= 1° 58' for the first deflec- 
- tion. For each subsequent full chord 2° 27' additional is 
turned off on the transit as far as the line can be seen. 
Say that the line cannot be seen farther than Sta. 116; the 
several deflections up to that point would be, for Sta. 112, 
1° 58'; Sta. 113, 4° 25'; Sta. 114, 6° 52'; Sta. 115, 9° 19'; Sta. 
116, 11° 46'. A hub and tack are driven at Sta. 116, and 
the transit moved up to that point or, what is better, 
to the P. T., if the station is visible from there. If the 
transit is set up at Sta. 116, the back-sight is taken on the 
P. C., with the limb clamped at 11° 46', as at the last 
observation. The telescope is then righted, and an addi¬ 
tional 11 °46' turned off for the new tangent, from which 
the subsequent deflections are turned off. For Station 
117 the deflection would be 11° 46' + 11° 46' -f- 2° 27' = 
25° 59'; for Sta. 118, 28° 26'; for Sta. 119, *30° 53'; for Sta. 
120, 33° 20'; for Sta. 121, 35° 47'. 

Before passing this point, we must know the length of 
the curve. As there are 48° 16)' total deflection, and each 
chord cuts off 4° 54' of it, it is evident there are as many 
100 foot chords as 4° 54' is contained in 48° 16'. deducing 

48.266 

the minutes to decimals and dividing, we have-= 

4.9 

9 A 5 chords for the length of the curve. This added to 
m b 19.8 gives us 121 + 04.8 for the end of the curve, 
and 04.8 feet for the last partial chord. We find the 



CURVELINEAR SURVEYING. 159 

deflection for this distance to be .07 , giving for the last 
deflection 35° 47' - r .07' = 35° 54'. 

The work should now prove itself, by coming out at 
the stake which was previously set for the end of the 
curve, and we may further test it by setting the transit 
up at the P. T., back-sight to Sta. 116, with the instrument 
clamped at 35° 54', as last used. Unclamp the limb and 
turn off as much more as has been turned from Sta. 116, 
35° 54' — 23° 32' = 12° 22', which added to 35° 54' = 48° 16', 
the total deflection. If the work has been accurately 
performed, a back-sight through the telescope should 
strike the P. I. It is very seldom that curves run in this 
way will come out just right, hence it is better to never 
set up the transit at points in the curve between the P. 0. 
and P. T. when it can readily be avoided. Still it is the 
ordinary and sometimes the only way the curves can be 
run. 

Let the student make the necessary calculations to 
locate curves from the following data: 

Ex. 2 .—Total deflection, 26° 50'. External (7/, Fig. 65), 
120.87 feet. P. C. at Sta. 112, -f 40. Transit moved every 
550 feet. 

Ex. 3 .—Total deflection, 35° 15'. External, 126.2 feet. 
P. I. at Sta. 262, + 07.3. T. P. at Sta. 263. 

Ex. 4 .—Total deflection, 18° 36'. Curve, 1° 2', P. I. at 
96, -f 42.6. T. P. at Sta. 93 and 100. 

The starting point of a curve is sometimes so situated 
that it is not convenient to set up the transit at that 
point, or to run the line from it if it were, as in streams, 
gullies, etc., and it then becomes convenient to set up 
the transit first at some intermediate point in the curve 
which has to be found. 

5. To Locate a Curve from the Middle 

Point.— Set the transit up at the P. I. Bisect the inte¬ 
rior angle bed (Fig. 65). Find the external of of the 
desired curve and measure it off on the line of bisection. 
This gives the middle point of the curve. The transit is 
then set up at this point and a back-sight taken either on 
the P. 0. or P. I., and the curve run in. Let the student 
make the necessary calculations and give the various 


160 


A MANUAL OF LAND SURVEYING. 


deflections which would be used on the transit to locate 
from the middle point the curve required in Ex. 1, Fig. 
133, the first back-sight to be taken from the P. C. Give 
the same, the back-sight being taken from the P. I. Also, 
solve the following curves, to be run from the middle 
points, back-sights from P. C., also from P. I.: 

Examples—1. Total deflection, 16° 24'. Curve, 1° 32'. 
P. I. at 96, -f- 27. 

2- Total deflection, 26° 18'. Curve, 2° 24'. P. I. at 13, 
+ fi 2.7. 

3. Total deflection, 35° 40'. Curve, 3° 16'. P. I. at 97, 
-f 62.6. 

It is sometimes convenient, from various reasons— 

6. To Locate the Curve with the Tran¬ 
sit at some other Intermediate Point on the 
Curve than the middle. Such points may be located 
by ordinates from the tangent. This is usually done to 
avoid obstacles in the line of the curve. To find approx¬ 
imately on the ground at what point the transit may 
be set up, the following formula may be used: 

Let x = length of the ordinate, 

d = distance along the tangent from the P. C., 
t = nat. tangent of % the deflection angle of the 
curve, 

Then x = dH. 

Example— To find whether the transit can be set up at 
a point on a 4° curve opposite a point on the tangent 4( 0 
feet from the P. C. 

t = nat. tang., 2° = 03.5. d‘ z = 16. .\ x — 56. A meas¬ 
ure of 56 feet from the tangent will show whether the 
transit can be set up at this point or not. 

It will be fonnd the most convenient in running the 
curve to select the point at a regular station at the end of 
a full chord, which may be located as follows: 

Example 1 .—Total deflection, 48° 48'. P. 1. at 62, -f 36. 
Curve, 4° . To find the 4th full station on the line of the 
curve, and locate the remainder of the curve from that 
point. 


CURVELINEAR SURVEYING. 


161 



First find the number 
of the station at the 
P. C. be — tangt. of 1° 


2599.2 


-4- 4 =-= 649.8 or 


4 


Fig. gg. 


6 + 49.8. This taken 
from 62 + 36 = 55 + 
86.2, which is the num¬ 
ber of the station at 
the P. C. From here 
to the 4th full station 
there is then a short 


chord of 13.8 feet and four full chords. The tangential 
angle cbd is therefore 4.138 X 2° = 8° 16%'; whence the 
deflection angle = 16° 33', the chord of which, bd, = 413.4. 
In the right-angled triangle bed , we now have the side bd 
= 413.4, and the angle cbd = 8° 16X', to find the sides be 
and cd , from which we find that be = 409.1 and cd — 58.5. 
The point c may be found by measuring from the P. I. 
649.8 — 409 = 240.7 = ec. Having thus located the point 
d, which is Station 60 on the curve, the transit is set up at 
that point, with the vernier clamped at 90° , and a back¬ 
sight taken to the point c. The upper clamp is then 
loosened and the limb brought to 16° 33', which gives the 
tangent from which the remainder of the curve is located. 

Let the student calculate the following curves: 

2. Total deflection, 36 20', P. I. at 26, -j- 44.6. Curve, 
2° 30", to be located from the 3rd full station on the curve. 

3. Total deflection, 61° 18'. P. I. at 42, + 28.5. Curve, 
4° 40', to be located from 6th full station on the curve. 

4. Total deflection, 42° 50'. P. I. at 112, + 72. Curve, 
3° 18', to be located from Station 114 + 50 on the curve. 

7 . Short Curves.— When the deflections between 
the lines are but small, and it is not important that any 
particular degree of curvature be used, it will be found 
convenient to make the curve an even two or four sta¬ 
tions in length. In case this is done, the curve may be 
marked out before the transit is moved from the P. I., 
after observing the deflection angle, and it will not be 


12 







162 


A MANUAL OF LAND SURVEYING. 


necessary to set it up on the curve at all. The middle of 
the curve will be located by laying off the external secant 
as before directed. The P. 0. and P. I. are also located as 
usual. If four stations are used, the intermediate sta¬ 
tions may be determined from the P. I., the same as if 
the transit were at the P. C. or P. T., the error being so 
small that it may usually be neglected. 

8- Passing Obstructions in the Line. —One 

method of doing this, by offset from the tangent, has 
already been sufficiently explained. Another method, 
which is very generally applicable, is by parallel offsets 
from the curve. An offset is made in any convenient 
direction far enough to pass the obstruction. The curve 
is continued from this point till the obstacle is passed, 
when the true line is regained by an inset equal to and 
parallel with the offset. If the lines are run in the man¬ 
ner indicated on page 82, (3), this will be a very simple 
matter, as the telescope will always point in the same 
direction when the verniers mark the same point oil 
the limb. 



Fig. 67 illustrates this method 
of passing obstacles, be and de 
are equal and parallel. 


c 


Fig. 67. 


9. Compound Curves, being a combination of 
simple curves, have their several components located in 
the same manner. They are usually run to fit the topog¬ 
raphy of the country through which they are laid, in order 
to get uniform gradients on street or railroad lines, or 
save labor and expense in construction. 










163 


CURVELINEAR SURVEYING}. 

Having the several straight lines determined which are 
to form the tangents of the curve, it is only necessary to 
find the degrees of curvature of the several component 
curves, which are then located in the manner already 
described. Usually there will be found on the ground 
special reasons for selecting a particular radius for one of 
the component curves, which will thus dictate the radii 
of the rest. 



Example 1. —Let ac, ce, eg , gi and ik repre ent tangents 
of the curve, and bed, def, fgh and hik the angles of 
deflection. 

Let ce = 1370, eg = 1200, gi = 1000. 

Let bed = 92° , def — 36° , fgh = 23° 15', and hik — 
43° 30', the corresponding curves of which we will-number 
1, 2, 3 and 4. 

Let the tangents ce and eg be united by a 3° curve. 

Required the radii or degrees of curvature of the 
remaining components of the curve, and the length of 
the curve. 

Suggestions. —First find the tangent of a 3° curve for 
an angle of 36°. Tangent of 1° curve for 36° = 1861.8; 
.*. for 3° curve = 620.6. This leaves 1370 — 620.6 = 749.4, 
length of tangent of curve No. 1. Tangent of 1° curve 
for 92° — 5933.2, which divided by 749.4 = 7.917° or 7° 55', 
the degree of curvature. Radius, 724.3. Length of curve 
92 

=-= 1162 feet. We find the tangent of curve No, 3 

7.917 







164 


A MANUAL OF LAND SURVEYING. 


by subtracting the tangent of curve No. 2, 620.6, from the 
length of the line eg , 1200, = 579.4. The tangent of a 1° 
curve for a deliection of 23° 15 J we find from the table 
to be 1178.8, which divided by 579.4 gives the degree of 
curve to be used, 2.034° = 2° 027 The calculations for the 
remainder of the curve are made in a similar manner. 

It is customary in running long lines for drains, rail¬ 
ways, etc., to run preliminary lines by angles, omitting 
the curves, till the location of the tangents is definitely 
determined. Stakes are set and numbered the same as on 
the final location. Both the staking and measuring are 
sometimes omitted, the lines being run as simple picket 
lines. In such case, when the final location is made, the 
line is staked out to the point of intersection of the 
tangents and afterward, as the curve is run in, the stakes 
between the P. C. and P. I. are taken up and moved to 
their proper place in the line of the curve. 

Examples for solution. —1. Let the student calculate 
the curves and plat the line from the following notes ©f a 
preliminary angle line, making all the calculations that 
would be required in the field, and giving the corrected 
numbers of the stations at the several P. C.’s, P. T.’s, P. 
C. C.’s and P. II. C.’s:— 


176 o 


165 o 

P. I, Compound Curve. Angle right, 14°. 

163 + 20 o 

P. I. Reverse Curve. Angle right, 36°. 

153 o 

P. I. Reverse Curve. Ang’e left. 17° 26'. 

144 + 26 o 

P. I. 2° Curve. Angle right, 16° 30' 

129 o 


118 o 


o 

oo 

o 


98 + 15 o 

P. I. of Reverse Curve. Angle right, 53° 12'. 

85 + 60 o 

P. I. Compound Curve. Angle left, 16°. 

76 + 48 o 

P. I. Compound Curve. Angle left, 7° 8'. 

67 o 

P. I.-Curve. Angle left, 12°23'. 

58 o 


o 

QO 


41 o 

P. 1. 2° 16' Curve. Angle right, 14°. 

30 o 


21 o 

P. I. 3° Curve. Angle left, 26° 32'. 

H o 


0 o 

N. 45° E. 










CURVELINEAR SURVEYING. 


165 


2. The following are notes of the north side of a street in 
Park Beidler. The measures are taken with a 6(5 foot 
tape of 100 links. The street is one chain wide. A tier 
of lots two chains deep is laid out on each side of the 
street. The lots are one chain wide on the street, and are 
marked by stakes set and numbered at regular intervals 
of one chain. The lines for the south side of the street 
and for the back ends of the two tiers of lots are to -be 
run with the transit and tape. Required the details of 
these lines and the widths of lots at the back end, the 
lot lines being at light angles with the street and on the 
radii of the curves. 


48 

45 

40 

36 

32 

26 

24 

21 

18 

12 

8 + 50 
6 + 20 
2 
0 


Intersect west line of Dawn Street. Course N. and S. 

P. T. 

P. R. C. 10° Curve right? 

P. C. C. 8° Curve left. 

P. C. C. 5° Curve left. 

P.C. 2° Curve left. 

P. T. 

P. C. C. 6° Curve right. 

P. R. C. 4° Curve right. 

P. C. C. 4° Curve left. 

P. R. C. 8° Curve left. 

P. C. C. 4° 30' Curve right. 

P. C. 3° Curve right. 

East at right angles with Sylvan St. Course N. and S. 


The following formula has been found very useful in 
solving many problems in the location of curves. Like 
the formula x = d 2 t in Art. 6, it is designed to express 
the length of an ordinate from the tangent to the curve: 

Let x = length of the ordinate, 

n — length of the curve in chords of 100 feet each, 
d = degree of curvature. 

Then x = %n 2 d. Thus a 6° curve will have diverged 
from its tangent at the end of 500 feet, 1X 5 2 X 6 = 131.25 
feet. 

By making d equal the difference of the degree of 
curvature of two curves of different radii but having a 
common origin, a? will be their divergence from each other 






166 A MANUAL OF LAND SURVEYING. 

at the end of n stations. This formula is not mathemat¬ 
ically exact, and therefore gives only approximate results; 
but it is sufficiently correct for all ordinary cases. It is 
easily remembered; it requires no tables; and with its aid, 
with such modifications as a little ingenuity will suggest, 
and a table of actual tangents for a 1° curve, the surveyor 
can solve almost any case that will ordinarily arise in the 
field. For example: Suppose a 5° curve to the right 8 
stations long has been located, and its extremity falls 28 
feet too far to the right to throw the tangent on the best 
ground. Making x = 28, we obtain d = \, showing that 
a 4° 30' curve starting from the same origin would pass 
through the required spot. Again: Suppose that in this 
same case the new curve is to commence 200 feet back of 
the first one; then the required divergence from the tan¬ 
gent will be | X 8 2 X 5 — 28 = 252. Substituting this 
value for x , and making n ==? 8 -f- 2, we have d — 2.88 = 
2 ° 53 ', 


ORIGINAL SURVEYS. 


167 


CHAPTER VIII. 

ORIGINAL SURVEYS. 

1. In land surveying, the surveyor has two distinct 
classes of problems to deal with. In the first class, he is 
called upon 

(а) To lay down upon the ground the corners and 
boundary lines of tracts of land of specified dimensions; 
and 

(б) To find the areas of tracts which are already defined 
by natural or artificial boundaries. 

In this class is included the original marking out upon 
the ground of the boundaries of every tract of land how¬ 
ever great or small. Hence we call surveys of this nature 
Original Surveys. 

2. When the boundaries have once been laid down 
upon the ground and marked by persons having authority 
to do so, then the surveyor, who is afterward called upon, 
has a different class of problems to deal with. He then 
has 

(а) To find the corner posts and monuments; 

(б) To re-locate them when lost; and 

(c) To retrace old boundary lines. 

Surveys of this nature we shall call Resurveys. 

3. Original Surveys include: First. The rectangu¬ 
lar surveys of the United States, known as the govern¬ 
ment survey; similar surveys in Canada and other coun¬ 
tries by government authority, and the subdivision of 
sections. Second. Surveys made by the proprietors in 
those regions where the government surveys do not ex¬ 
tend, including in the United States the surveys of all 


168 


A MANUAL OF LAND SURVEYING. 


land not granted by the original states of the Union to 
the general government; and surveys for town plats, 
highways and like purposes. 

4 United States Survey.—The territory embraced 
within the present States of Ohio, Indiana, Illinois, Mich¬ 
igan, Wisconsin, and Tennessee, that part of Minnesota 
lying east of the Mississippi River, and all of Alabama 
and Mississippi lying north of the thirty-first parallel, 
was held by Massachusetts, Connecticut, New York, Vir¬ 
ginia, North Carolina, South Carolina, and Georgia, under 
grants from Great Britain, during their colonial condi¬ 
tion. These territorial interests were surrendered to the 
General Government of the Union by the last named 
States at different times hereinafter set forth, and consti¬ 
tuted the nucleus of our public domain with some reser¬ 
vations as to former grants, and was the remainder of the 
territory conceded to the United States under the defini¬ 
tive treaty of 1783, and consisted of 401,955.91 square 
miles, or 259,171,787 acres. This was the public domain 
of the United States on April 30,1803, the date of the 
Louisiana purchase, and for which the original survey 
and disposition laws were made. 

The United States were recognized by the Crown in the 
definitive treaty of peace with Great Britain as “free 
sovereign and independent States, and that he treats with 
them as such, and for himself, his heirs, and successors 
relinquishes all claims to the government, proprietary 
and territorial rights of the same, and every part thereof.” 

The Government of the United States acquired as cus¬ 
todian for the Nation, lands known as the public domain 
as follows: 

From States (colonies prior to July 4,1776) ceded under 
the Confederation and under the Constitution. 

This was in pursuance of a resolution of the Congress 
of the Confederation passed Tuesday, October 10, 1780, 
providing for the reception and care of such unappropri- 


ORIGINAL SURVEYS. 


109 


ated lands as might be ceded by States to the United 
states, and for the disposition of the same for the com¬ 
mon benefit of the United States. 

The dates of cession of these lands to the United States 
were as follows: 


Colony. 

State. 

Date of Cession. 

New Hampshire. 

New Hampshire. 

No cession. 

New York.... 

Rhode Island and Provi- 

New York. 

March 1,1781. 

dence Plantations.. 

Rhode Island. 

No cession. 

New Jersey. 

New Castle, Kent and Sus- 

New Jersey. 

Do. 

sex. on Delaware. 

Delaware. 

Do. 

Pennsylvania... 

Pennsylvania. 

Do. 

Virginia.. 

Virginia. 

March l, 1784, and De¬ 
cember 30, 1788.* 

Maryland. 

Maryland. 

No cession. 

Massachusetts Bay. 

Massachusetts. 

April 19, 1785. 

Connecticut. 

Connecticut. 

September 13,1786; con¬ 
firmed May 30,1800. 

South Carolina. 

South Carolina. 

August 9, 1787. 

North Carolina... 

North Carolina. 

February 25,1790. 

Georgia. 

Georgia. 

April 24, 1802. 


*An act to change the conditions of the cession of March 1,1784, only 
so far as to ratify the fifth article of the compact of the ordinance of 
1787. 

AREA OF CESSIONS. 



Sq. miles. 

Acres. 

Massachusetts (disputed) claimed (estimated)* 

51,000.00 

34,560,000 

Connecticu (disputed) and Western Reserve 
and Fire-lands (estimated)*. 

40.000.00 

25,600,000 

From New York and Massachusetts cession, 
actual.. 

315.91 

202,187 

169,959,680 

From Virginia (disputed and undisputed) to 
the United States (exclusive of Kentucky and 
including area of Western Reserve and the 
Fire-lands)f. 

265,562.00 

South Carolina cession. 

4,900.00 

3,136,000 

North Carolina cession, nominal, because the 
area of Tennessee was almost covered with 
reservations. 

45,600.00 

29,184,000 

Georgia cession. 

88,578.00 

56.689,920 

Total actual State cessions to the United 
States for public domain. 

404,955.91 

259,171,787 


♦The area above was also claimed by Virginia and included in her 
cession. 

fConnecticut’s jurisdictional cession of the Western Reserve and 
Fire-lands, containing about 3,800,000, included under Virginia cession. 























































170 A MANUAL OF LAND SURVEYING. 


AREA OF PURCHASES—PUBLIC AND NATIONAL DOMAIN. 



Sq. miles. 

Acres. 

Louisiana, piirehase April 30, 1803. 

1,182,752 

59,268 

522,568 

756,961,280 

East and West Florida, Feb. 22, 1819. 

flan dal ii no Hidaleo Febrna.rv 2 1848. 

37,931,520 

334,443,520 

61,892,480 

29,142,400 

369,520,600 

State of Texas, November 25, 1850. 

96,707 

45,535 

577,390 

Gadsden piirehase, Deeember 30, 1853. 

Alaska.pnreba.se Mareb 30, 1807 . 



2,484,220 

1,589,900,800 


At a total cost of $88,157,389.98. 

The Texas annexation of 1845 added to the national 
domain the area of the present State of Texas, viz., 
274,356 square miles, or 175,587,840 acres, included in the 
national domain, besides the purchase of 1850 from the 
State, now public domain. 

The total area of purchased and annexed territory, in¬ 
cluded in the national an-d public domain since 1803, is 
2,758,576 square miles, or 1,765,488,640 acres, at a total cost 
of $S8,157,389.98 for the purchase, and including’ the 
Georgia cession of 1802, $6,200,000. 

5. The present system of survey of the public 
lands was inaugurated by a committee appointed by the 
Continental Congress, and consisting of the following 
delegates: Hon. Thomas Jefferson, chairman, Virginia; 
Hon. Hugh Williamson, North Carolina ; Hon. David 
Howell, Rhode Island, Hon. Ethridge Gerry, Massachu¬ 
setts; Hon. Jacob Read, South Carolina. 

On the 7th of May, 1784, this committee reported “An 
ordinance for ascertaining the mode of locating and dis¬ 
posing of lands in the western territory, and for other 
purposes therein mentioned.” This ordinance required 
the public lands to be divided into “hundreds” of ten 
geographical miles square, and those again to be sub¬ 
divided into lots of one mile square each, to be numbered 
from 1 to 100, commencing in the north-western corner, 
and continuing from west to east and from east to west 






















ORIGINAL SURVEYS. 


171 


consecutively. This ordinance was considered, debated, 
and amended, and reported to Congress April 26,1785, and 
required the surveyors “ to divide the said territory into 
townships of 7 miles square, by lines running due north 
and south, and others crossing these at right angles. * * * 
The plats of the townships, respectively, shall be marked 
by subdivisions into sections of 1 mile square, or 640 
acres, in the same direction as the external lines, and 
numbered from 1 to 49. * * * And these sections shall be 
subdivided into lots of 320 acres.” 

This is the first record of the use of the terms “ town¬ 
ship ” and “ section.” 

May 3, 1785, on motion of Hon. William Grayson, of 
Virginia, seconded by lion. James Monroe, of Virginia, 
the section respecting the extent of townships was 
amended by striking out the words “seven miles square” 
and substituting the words “ six miles square.” The rec¬ 
ords of these early sessions of Congress are not very full 
or complete; but it does not seem to have occurred to the 
members until the 6th of May, 1785, that a township six 
miles square could not contain 49 sections of 1 mile square. 
At that date a motion to amend was made, which pro¬ 
vided, among other changes, that a township should con¬ 
tain 36 sections; and the amendment was lost. The or¬ 
dinance as finally passed, however, on the 20th of May, 
1785, provided for townships 6 miles square, containing 
36 sections of 1 mile square. The first public surveys 
were made under this ordinance. The townships, 6 miles 
square, were laid out in ranges, extending northward 
from the Ohio River, the townships being numbered from 
south to north, and the ranges from east to west. The 
region embraced by the surveys under this law forms a 
part of the present State of Ohio, and is usually styled 
“The Seven Ranges.” In these initial surveys only the 
exterior lines of the townships were surveyed, but the 
plats were marked by subdivisions into sections of 1 mile 
square, and mile corners were established on the town- 


172 A MANUAL OF LAND SURVEYING. 

ship lines. The sections were numbered from 1 to 36, 
commencing 1 with No. 1 in the southeast corner of the 
township, and running from south to north in each tier 
to No. 36 in the northwest corner of the township, as 
shown in the following diagram: 


36 

30 

24 

18 

12 

6 

35 

20 

23 

17 

11 

5 

34 

28 

22 

16 

10 

4 

33 

27 

21 

15 

9 

3 

32 

26 

20 

14 

8 

2 

31 

25 

19 

13 

7 

1 






The surveys were made under the direction of the 
Geographer of the United States. 

The act of Congress approved May 18,1796, provided 
for the appointment of a surveyor-general, and directed 
the survey of the lands northwest of the Ohio River,and 
above the mouth of the Kentucky River, “ in which the 
titles of the Indian tribes have been extinguished.” Un¬ 
der this law one-half of the townships surveyed were 
subdivided into sections “ by running through the same, 
each way, parallel lines at the end of every two miles, 
and by making a corner on each of said lines at the end 
of every mile,” and it further provided that “ the sections 
shall be numbered, respectively, beginning with the num¬ 
ber one in the northeast section and proceeding west and 
east alternately, through the township, with progressive 
numbers till the thirty-sixth be completed.” This method 
























































ORIGINAL SURVEYS. 173 

of numbering 1 sections, as shown by the following dia¬ 
gram, is still in use : 


6 

5 

4 

3 

2 

1 

7 

8 

9 

10 

11 

12 

18 

17 

16 

15 

11 

13’ 

19 

20 

21 

22 

23 

24 

30 

29 

28 

27 

2G 

25 

31 

32 

33 

34 

35 

36 


The act of Congress approved May 10, 1800, required 
the “townships west of the Muskingum, which * * * are 
directed to be sold in quarter townships, to be subdivided 
< into half sections of three hundred and twenty acres each, 
as nearly as may be, by running parallel lines through the 
same from east to west, and from south to north, at the 
distance of one mile from each other, and marking cor¬ 
ners, at the distance of each half mile on the lines run¬ 
ning from east to west, and at the distance of each mile 
on those running from south to north. * * * And the 
interior lines of townships intersected by the Muskingum, 
and of all the townships lying east of that river, which 
have not been heretofore actually subdivided into sec¬ 
tions, shall also be run and marked. * * * And in all 
cases where the exterior lines of the townships thus to be 
subdivided into sections or half sections shall exceed, or 
shall not extend, six miles, the excess or deficiency shall 
be specially noted, and added to or deducted from the 
western and northern ranges of sections or half sections 
in such township, according as the error may be in run¬ 
ning the lines from east to west or from south to north.” 


























































174 


A MANUAL OF LAND SURVEYING. 


6. The acts of Congress defining the system of public 
land surveys, and the principles to be employed in carry¬ 
ing them out, are to be found in the United States Stat¬ 
utes as follows: 


Act of May 

18, 1796, 

Volume 

1, 

Chap. 

29. 

44 

44 

10, 1800, 

44 

2, 

44 

55. 

a 

Feb. 

11, 1805, 

44 

0 

44 

11. 

u 

April 

24,1820, 

44 

3, 

44 

51. 

44 

44 

5, 1832, 

44 

4, 

44 

55. 

u 

May 

30, 1862, 

44 

12, 

44 

86. 

u 

March 

3, 1875, 

44 

18, 

44 

130. 

44 

44 

3, 1875, 

44 

10, 

44 

105. 


Such portions of the various acts as are now in force 
are published by the government in a volume entitled 
“ Existing Land Laws.” Those Sections which refer di¬ 
rectly to the surveys are as follows: 

7. United States Laws relating- to Surveys and 
Surveyors.— Sec. 77. There shall be appointed by the 
President, by and with the advice and consent of the 
Senate, a surveyor-general for the States and Territories 
herein named, embracing, respectively, one surveying 
district, namely: Louisiana, Florida, Minnesota, Kansas, 
California, Nevada, Oregon, Nebraska and Iowa, Dakota, 
Colorado, New Mexico, Idaho, Washington, Montana, 
Utah, Wyoming, Arizona. 

3 Stat. 755; 4 id. 492; 9 id. 496; 10 id. 244, 30C, 308, 309, 611; 11 id, 212; 
12 id, 176, 211, 214; 14 id. 77, 85. 314, 542; 15 id. 91; 16 id. 65, 240; 17 
id. 76; 18 id. 18, 34, 121,122, 123, 201, 303; 19 id. 126, 207i R. S. 2207. 

Sec. 84, Every surveyor-general shall, before entering 
on the duties of his office, execute and deliver to the Sec¬ 
retary of the Interior a bond, with good and sufficient 
security, for the penal sum of thirty thousand dollars, 
conditioned for the faithful disbursement, according to 
law, of all public money placed in his hands, and for the 
faithful performance of the duties of his office; and the 
President has discretionary authority to require a new 





ORIGINAL SURVEYS. 


175 


bond and additional security, under the direction of the 
Secretary of the Interior, for the lawful disbursement of 
public moneys. 

3 Stat. 697; R. S. 2215, 2216, U. S. v. Vanzandt, 11 Wheat, 184; U. S. v. 
Tingey, 5 Pet. 115; Farrar and Brown v. U. S„ 5 id. 373; U. S. v. 
Bradley, 10 id. 343; U. S. vs. Linn, 15 id. 290; U. S. v. Prescott, 3 
How. 578; U. S. v, Boyd, 5 id. 29; Bryan v. U. S., 1 Black, 140; Bov 
den v. United States, 13 Wall. 17; Bevans v. U. S , 13-id. 50; U. S. 
v. Thomas, 15 id. 337; U. S. v. Stephenson, 1 McClean, C. C. 462; 
U. S. v. Linn, 2 id. 501; U. S. v. Ward, 3 id. 179. 8 Op. Att. Gen. 7. 
Cir. G. L. O., July 1,1871; id. May 14,1879. Treasury Cir., July 13, 
1871 (Copp’s L. L. 783; 1 Lester’s L. L. 312, 314). 

Sec. 85. The commission of each surveyor-general shall 
cease and expire in four years from the date thereof, un¬ 
less sooner vacated by death, resignation, or removal 
from office. 

3 Stat. 697; R. S. 2217. Best v. Polk, 18 Wall. 112. Decision Com. 
G. L. O., Feb. 20,1858 (1 Lester’s L. L. 340). 

Sec. 86. Every surveyor-general, except where the Pres¬ 
ident sees cause otherwise to determine, is authorized to 
continue in the uninterrupted discharge of his regular 
official duties after the day of expiration of his commis¬ 
sion and until a new commission is issued to him for the 
same office, or until the day when a successor enters upon 
the duties of such office; and the existing official bond of 
any officer so acting shall be deemed good and sufficient 
and in force until the date of the approval of a new bond 
to be given by him, if recommissioned, or otherwise, for 
the additional time he may so continue officially to act, 
pursuant to the authority of this section.. 

10 stat. 247 ; 18 id, 62; R. S. 2222. 

Sec. 87. Whenever the surveys and records of any sur¬ 
veying district are completed, the surveyor-general thereof 
shall be required to deliver over to the Secretary of State 
of the respective states, including such surveys, or to 
such other officer as may be authorized to receive them, 
all the field-notes, maps, records, and other papers apper¬ 
taining to land titles within the same; and the office of 


176 


A MANUAL OF LAND SURVEYING. 


surveyor-general in every such district shall thereafter 
cease and be discontinued. 

5 stat. 384; 19 id. 121; R. S. 2218. 

Sec. 88. In all cases of discontinuance, as provided in 
the preceding section, the authority, powers, and duties 
of the surveyor-general in relation to the survey, resur¬ 
vey, or subdivision of the lands therein, and all matters 
and things connected therewith, shall be vested in#md 
devolved upon the Commissioner of the General Land 
Office. 

10 stat. 152; R. S. 2219. 

Sec. 89. Under the authority and direction of the Com¬ 
missioner of the General Land Office, any deputy surveyor 
or other agent of the United States shall have free access 
to any such field-notes, maps, records, and other papers 
for the purpose of taking extracts therefrom or making 
copies thereof without charge of any kind; but no transfer 
of such public records shall be made to the authorities of 
any State until such State has provided by law for the 
reception and safe-keeping of such public records and for 
the allowance of free access thereto by the authorities of 
the United States. 

10 Stat. 152; 18 id. 62; R. S. 2220, 2221. 

Sec. 90. Every surveyor-general shall engage a sufficient 
number of skillful surveyors as his deputies, to whom 
he is authorized to administer the necessary oaths upon 
their appointments. He shall have authority to frame 
regulations for their direction, not inconsistent with law 
or the instructions of the General Land Office, and 'to 
remove them for negligence or misconduct in office. 

Taylor and Quarlls v. Brown, 5 Cranch, 234; Craig et al.v. Braxford, 
3 Wheat, 594; Ellieott ct al. v. Pearl, 10 Pet. 412; Brown’s Lessee 
v. Clements, 3 How. 650. Reed v. Conway 20 Mo. 22; same case, 
26 id, 13; Hamil v. Carr, 21 Ohio St. 258; Doe v. Hildreth, 2 Ind. 
274; McClintock v. Rodgers, ll Ills. 279. Cir. G. L, O., June 26, 
1880. 

Second. He shall cause to be surveyed, measured, and 
marked, without delay, all base and meridian lines through 





ORIGINAL SURVEYS. 


177 


such points and perpetuated by such monuments, and 
such other correction parallels and meridians as may be 
prescribed by law or by instructions from the General 
Land Oflice in respect to the public lands within his sur¬ 
veying district, to which the Indian title has been or may 
be hereafter extinguished. 

Gazzen v. Phillips’ Lessee, 20 How. 372. 3 Op. Att. Gen., 281, 284. 
Atsliire v. Hulse, 1 Ohio, 170; Hastings v. Stevenson, 2 d. 9; Mc¬ 
Kinney v. McKinney, 8 id. 423; Ilamil v. Carr, 21 Ohio St. 258; 
Hendrick v. Eno, 42 Iowa 411; Saint Louis v. Walker, 40 Mo. 383; 
Jordan v. Barrett, 13 La. 24; Fowler v. Duval, 11 id. 501; Cox v. 
Jones, 47 Cal. 412. Cir. G. L. O., June 20,1880. 

Third. He shall cause to be surveyed all private land 
claims within his district after they have been confirmed 
by authority of Congress, so far as may be necessary to 
complete the survey of the public lands. 

Menard’s Heirs v. Massey, 8 How. 293; Kissell v. St. Louis Public 
Schools, 18 id. 19; Stanford v. Taylor, 18 id. 409; Ballanee v. For¬ 
syth, 24 id. 183; U. S. v. Fossat, 25 id. 445; Carondelet v. St. Louis, 
1 Black, 179; U. S. v. Sepulveda, 1 Wall. 104; U. S. v. Ilalleck, 1 id. 
439; U. S. v. Billings, 2 id. 444; Sutter’s case, 2 id. 502; U. S. v. 
Pacheco, 2 id. 587; Fossat case, 2 id. 649; Dehon v. Bernal, 2 id. 
774; U. S. v. Armijo, 5 id. 444; Higueras v. U. S. 5 id. 827; Maguire 
v. Tyler, 8 id. 650; Lynch, v. Bernal 9 id. 315; Henshaw v. Bissell, 
18 id. 255; Shepley et al. v. Cowan et al., 1 Otto, 330; Miller et al. v # 
Dale et al., 2 id. 473; Van Reynegand v. Bolton, 5 id. 33; U. S. v. 
Throckmorton, 8 id. 61; Snyder v, Sickles, 8 id. 203; Scull v. U. S., 
8 id, 410. Bissell v. Henshaw, 1 Saw. C. C. 553; Leroy v. Jamison, 
3 id. 369. Gibson v. Chouteau, 39 Mo. 536; Milburn v. Hardy, 28 id. 
514; Funkliouser v. Ilantz, 29 id. 540; Dent v. Legesson, 29 id. 489; 
Carondelet v. St. Louis, 29 id. 527; Maguire v. Tyler, 30 id. 202; 
Robins v. Eckler, 36 id. 494; Clark v. Heammerle, 36 id. 620; Gib¬ 
son v. Chouteau, 39 id. 536; Vasquez v. Ewing, 42 id. 247; Glasgow 
v. Lindell,50 id. 60; Rector v. Gaines, 19 Ark. 70; Ashley v. Rector, 
20 id. 359; Meaux v. Breaux, 10 Martin (La.) 364; Moon u. Wilkin¬ 
son, 13 Cal. 478; Boggs v. Mining Co., 14 id. 279; Mott v. Smith, 16 
id. 534; Johnson v. Van Dyke, 20 id. 225; McGarrahan v. Maxwell, 
27 id. 75; Treadway v. Semple, 28 id. 652; Searle v. Ford, 29 id. 104; 
Mahoney v. Van Winkle, 33 id. 448; Morrill v. Chapman, 35 id. 85; 
Yates v. Smith, 38 id. 60; San Diego v. Allisqn,46 id. 163. De¬ 
cisions Sec. Int., July 16,1872; Aug. 8,1876; Aug. 17,1876; March 
16,1877. Decisions Com. G. L. O., Aug. 18, 1S60; Sept. 18, 1874; 
Nov. 3, 1874; Sept. 18, 1875; Oct. 28, 1875; June 26, 1879. Cir. G. L. 
O., June 26,1880. 


13 




178 


A MANUAL OF LAND SURVEYING. 


Fourth. He shall transmit to the register of the respec¬ 
tive land offices within his district general and particular 
plats of all lands surveyed by him for each land district; 
and he shall forward copies of such plats to the Commis¬ 
sioner of the General Land Office. 

Barnard v. Ashley, 18 How. 43; Water and Mining Co. v. Bugbee,6 
Otto. 1G5; Hamil v. Carr, 21 Ohio St. 258; Doe v. Hildreth, 2 Ind 
274; Pope v. Athearn, 42 Cal. 606; Com. G. L. O. Instructions to 
Surveyor-General, April 17,1879. 

Fifth. He shall, so far as is compatible with the desk 
duties of his office, occasionally inspect the surveying 
operations while in progress in the field, sufficiently to 
satisfy himself of the fidelity of the execution of the 
work according to contract, and the actual and necessary 
expenses incurred by him while so engaged shall be 
allowed; and where it is incompatible with his other duties 
for a surveyor-general to devote the time necessary to 
make a personal inspection of the work in progress, then 
he is authorized to depute a confidential agent to make 
such examination, and the actual and necessary expenses 
of such person shall be allowed and paid for that service, 
and five dollars a day during the examination in the field; 
but such examination shall not be protracted beyond 
thirty days, and in no case longer than is actually neces - 
sary; and when a surveyor-general, or any person em¬ 
ployed in his office at a regular salary, is engaged in such 
special service he shall receive only his necessary expenses 
in addition to his regular salary. 

1 stat. 464; 13 id. 325 ; 4 id. 492; 10 id. 245, 247; 18 id. 34; 19 id. 126; R. 
S. 2223. Sec. Int. Instructions, July 1, 1874; Sept. 21, 1874. Cir. 
G. L. O., June 26,1880. 

Sec. 91. Every deputy surveyor shall enter into a bond, 
with sufficient security, for the faithful performance of 
all surveying contracts confided to him: and the penalty 
of the bond, in each case, shall be double the estimated 
amount of money accruing under such contracts, at the 
rate per mile stipulated to be paid therein. The suffici- 



ORIGINAL SURVEYS. 179 

eney of the sureties to all such bonds shall be approved 
and certified by the proper surveyor-general. 

4Stat. 493; 10 id. 247; R. S. 2230. U. S. v. Vanzandt, 11 Wheat. 184; 
U. S. v. Tingey, 5 Pet. 115; Farrar et al. v. U. S., 5 id. 373; U. S. v. 
Bradley, 10 id. 343; U. S. v. Linn, 15 id. 290. U. S. v. Stephenson, 
l McLean, C C. 462. 

Sec. 92. The surveyors-general, in addition to the oath 
now authorized by law to be administered to deputies on 
their appointment to office, shall require each of their 
deputies, on the return of his surveys, to take and sub¬ 
scribe an oath that those surveys have been faithfully 
. and correctly executed according to law and the instruc¬ 
tions of the surveyor-general. 

9 Stat. 79; R. S. 2231. Ellicott and Meredith v. Pearle, 10 Pet. 412; 
U. S. v. Hanson, 16 id. 196; Bollard et al. v. Dwight et al.. 4 Cranch, 
421; Taylor et al. v. Brown. 5 id. 234. Cir. G. L. O., June 26,1880. 

Sec. 93. The district attorney of the United States, in 
whose district any false, erroneous, or fraudulent surveys 
have been executed, shall, upon the application of the 
proper surveyor-general, immediately institute suit upon 
the bond of such deputy, and the institution of such suit 
shall act as a lien upon any property owned or held by 
such deputy or his sureties at the time such suit was 
instituted. 

9 Stat. 79; R. S. 2232. 

Sec. 99, The public lands shall be divided by north and 
south lines run according to the true meridian, and by 
others crossing them at right angles, so as to form town¬ 
ships of six miles square, unless where the line of an 
Indian reservation, or of tracts of land heretofore sur¬ 
veyed or patented, or the course of navigable rivers, may 
render this impracticable; and in that case this rule must 
be departed from no further than such particular circum¬ 
stances require. 

McKinney v, McKinney, 8 Ohio, 423; Hamil v. Carr, 21 Ohio St. 258, 
Decision Sec. Int, Jan. 24,1880. Cir. G. L. O , June 26,1880. 

Second. The corners of the townships must be marked 
with progressive numbers from the beginning, each dis- 


180 


A MANUAL OF LAND SURVEYING. 


tance of a mile between such corners must be also dis¬ 
tinctly marked with marks different from those of the 
corners. 

Third. The township shall be subdivided into sections, 
containing, as nearly as may be, six hundred and forty 
acres each, by running through the same, each way, par¬ 
allel lines at the end of every two miles; and by making 
a corner on each of such lines, at the end of every mile. 
The sections shall be numbered, respectively, beginning 
with the number one in the northeast section and pro¬ 
ceeding west and east alternately through the township 
with progressive numbers till the thirty-six be completed. 

Grogan v. Knight, 27 Cal. 51G. Decision Sec. Int., April 14,1879. Cir. 
G. L. 0., June 26,1880. 

Fourth. The deputy surveyors, respectively, shall cause 
to be marked on a tree near each corner established in the 
manner described, and within the section, the number of 
such section, and over it the number of the township 
within which such section may be; and the deputy sur¬ 
veyors shall carefully note, in their respective field-books, 
the names of the corner-trees marked and the numbers 
so made. 

Cir. G. L. O., June 26, 1880. 

Fifth. Where the exterior lines of the townships which 
may be subdivided into sections or half-sections exceed, 
or do not extend six miles, the excess or deficiency shall 
be specially noted, and added to or deducted from the 
western and northern ranges of sections or half-sections 
in such townships, according as the error may be in run¬ 
ning the lines from east to west, or from north to south; 
the sections and half-sections bounded on the northern 
and western lines of such townships shall be sold as con¬ 
taining only the quantity expressed in the returns and 
plats respectively, and all others as containing the com¬ 
plete legal quantity. 

Knight v. Elliott, 57 Mo. 317; Vaughn v. Tate, 64 id. 491; Walters v. 
Commons, 2 Port. (Ala ) 38; Lewen v. Smith, 7 id 428. Decision 
Sec. Int., April 14,1879. Cir. G. L. O., June 26,1880. 


ORIGINAL SURVEYS. 


181 


Sixth. All lines shall be plainly marked upon trees, and 
measured with chains, containing two perches of sixteen 
and one-half feet each, subdivided into twenty-five equal 
links; and the chain shall be adjusted to a standard to be 
kept for that purpose. 

Bradley v. Taylor, 5 Cranch, 191; Mclvers v. Walker, 9 id. 173; Sliipp 
v. Miller’s Heirs, 2 Wheat. 316; Holmes v. Trout, 7 Pet. 171; Brown 
v. Huger, 21 How. 305; Meron v. Whitney, 5 Otto, 551; Robinson 
v. Moon, 4 McLean, C. C. 279. Oakley v. Stuart, 52 Cal. 521. Cir. 
G. L. O., June 26, 1880. 

Seventh. Every surveyor'shall note in his field-book the 
true situations of all mines, salt licks, salt springs, and 
and mill-seats which come to his knowledge; all water 
courses over which the line he runs may pass; and also 
the quality of the lands. 

Newsom v. Pryor’s Lessee, 7 Wheat. 7; Preston v. Bowman, 6 id. 580; 
Patterson v Jenks, 2 Pet. 216. 

Eighth. These field- books shall be returned to the sur¬ 
veyor-general, who shall cause therefrom a description of 
the whole lands surveyed to be made out and transmitted 
to the officers who may superintend the sales. He shall 
also cause a fair plat to be made of the townships and 
fractional parts of townships contained in the lands, de¬ 
scribing the subdivisions thereof and the marks of the 
corners. This plat shall be recorded in books to be kept 
for that purpose; and a copy thereof shall be kept open 
at the surveyor-general’s office for public information, 
and other copies shall be sent to the places of the sale 
and to tne General Land Office. 

1 Stat. 465; 2 id. 73; 19 id. 348; R. S. 2395. Taylor ct al. v. Brown, 5 
Crancli, 234; Barnard v. Asliley, 18 How. 43; Water and Mining 
Co. v. Bugbee, 6 Otto, 165. Rector v. Gaines, 19 Ark. 70; Lewen v. 
Smith, 5 Port. (Ala.) 428; Mott v. Smith, 16 Cal. 534; Hamil v. Carr, 
21 Ohio St. 258; Doe v. Hildreth, 2 Ind. 274; McClintockr. Rod¬ 
gers, ll Ills. 279. Decision Sec. Int., Jan. 15,1878 Decision Com. 
G. L. O., April 17, 1879. 

Sec. 100 . The boundaries and contents of the several 
sections, half-sections, and quarter-sections of the public 


182 


A MANUAL OP LAND SURVEYING. 


lands shall be ascertained in conformity with the follow¬ 
ing- principles: 

First. All the corners marked in the surveys, returned 
by the surveyor-general, shall be established as the proper 
corners of sections, or subdivisions of sections, which 
they were intended to designate; and the corners of half 
and quarter sections, not marked on the surveys, shall be 
placed as nearly as possible equidistant from those two 
corners which stand on the sqme line. 

Second. The boundary lines, actually run and marked 
in the surveys returned by the surveyor-general, shall be 
established as the proper boundary lines of the sections, 
or subdivisions, for which they were intended, and the 
length of such lines, as returned, shall be held and con¬ 
sidered as the true length thereof. And the boundary 
lines which have not been actually run and marked shall 
be ascertained by running straight lines from the estab¬ 
lished corners to the opposite corresponding corners; but 
in those portions of the fractional townships where no 
such opposite corresponding corners have been or can be 
fixed, the boundary lines shall be ascertained by running 
from the established corners due north and south or east 
and west lines, as the case may be, to the water-course, 
Indian boundary line, or other external boundary of such 
fractional township. 

Mott v. Smith, 1G Cal. 534; Guin v. Brandon, 29 Ohio St. 656; McClin- 
toclc v. Lodgers, 11 Ills. 279; Goodman v, Mvrick, 5 Greg. 65. Cir. 
G. L. O., June 26 ,1880. 

Third. Each section or subdivision of section, the con¬ 
tents whereof have been returned by the surveyor-gen¬ 
eral, shall be held and considered as containing the exact 
quantity expressed in such return; and the half-sections 
and quarter-sections, the contents whereof shall not have 
been thus returned, shall be held and considered as con¬ 
taining the one-half or the one-fourth part, respectively, 


ORIGINAL SURVEYS. 183 

of the returned contents of the section of which they 
make part. 

2 Stat. 313; R. S. 2396. Lindsey v. Hawes, 2 Black, 554; U. S. v. Pa¬ 

checo, 2 Wall. 587; Railway Co. v. Scliurmier, 7 id. 272; County of 
Saint Clair v. Livingston, 23 id. 46'; Heidekoper v. Brooms, 1 
Wash. C. C. 109; Coon v. Pen, 1 Pet. C. C. 496. 2 Op. Att. Gen. 
578. Knight v. Elliott, 57 Mo. 317; Vaughn v. Tate, 64 id. 491; 
Waters v. Commons, 2 Port. (Ala.) 38; Lewen v. Smith, 7 id. 428; 
Billingsly v. Bates, 30 Ala. 376; Doe v. Hildreth, 2 Ind. 274; Gro¬ 
gan v. Knight, 27 Cal. 516. Decision Com. G. L. O., May 17,1875. 
Cir. G. L. O., June 26, 1880. 

Sec. 101. In every case of the division of a quarter-sec¬ 
tion the line for the division thereof shall run north and 
south, and the corners and contents of half quarter-sec¬ 
tions which may thereafter be sold shall be ascertained 
in the manner and on the principles directed and pre¬ 
scribed by the section preceding, and fractional sections 
containing one hundred and sixty acres or upwards shall 
in like manner, as nearly as practicable, be subdivided 
into half quarter-sections, under such rules and regula¬ 
tions as may be prescribed by the Secretary of the Inte¬ 
rior, and in every case of a division of a half quarter- 
section, the line for the division thereof shall run east 
and west, and the corners and contents of quarter quarter- 
section, which may thereafter be sold, shall be ascertained, 
as nearly as may be, in the manner and on the principles 
directed and prescribed by the section preceding; and 
fractional sections containing fewer or more than one 
hundred and sixty acres shall in like manner, as nearly as 
may be practicable, be subdivided into quarter quarter- 
sections, under such rules and regulations as may be pre¬ 
scribed by the Secretary of the Interior. 

3 Stat. 566 ; 4 id. 503 ; R. S. 2397. Gazzam v. Phillips’ Lessee, 20 How. 

372; Railway Co. v. Scliurmier, 7 Wall. 272. Buel v. Tuley, 4 Mc¬ 
Lean, c. C. 268. Wharton v. Littlefield, 30 Ala. 245. 3 Op. Att. 
Gen. 281,284. Decision Sec. Int., April 14,1879. Decision Com. 
G. L, O., May 17,1875. Cir. G. L. O., June 26, 1880. 

Sec. 102. Whenever, in the opinion of the President, a 
departure from the ordinary method of surveying land 


m 


A MANUAL OF LAND SURVEYING. 


on ciny river, lake, bayou, or water-course would promote 
the public interest, he may direct the surveyor-general, 
in whose district such land is situated, and where the 
change is intended to be made, to cause the lands thus 
situated to be surveyed in tracts of two acres in width 
lionting on any river, bayou, lake, or water-course, and 
running back the depth of forty acres; which tracts of 
land so surveyed shall be offered for sale entire, instead 
of in half quarter-sections, and in the usual manner, and 
on the same terms in all respects as the other public lands 
of the United States. 

4 Stat. 34; R. S. 2407. 

Sec. 103. In extending the surveys of the public lands 
in the State of Nevada, the Secretary of the Interior may 
vary the lines of the subdivisions from a rectangular 
form, to suit the circumstances of the country. 

14 Stat. 8G; R. S. 2408. Heydenfeldt v. Mining Co., 3 Otto, 634. 

Sec. 104. The Secretary of the Interior, if he deems it 
advisable, is authorized to continue the surveys in Ore¬ 
gon and California, to be made after what is known as 
the geodetic method, under such regulations and upon 
such terms as have been or may hereafter be prescribed 
by the Commissioner of the General Land Office; but 
none other than township lines shall be run where the 
land is unfit for cultivation; nor shall any deputy sur¬ 
veyor charge for any line except such as may be actually 
run and marked or lor any line not necessary to be run. 

9 Stat. 496; 10 id. 245; li. S. 2409. , 

Sec. 105. Whenever, in the opinion of the Secretary of 
the Interior, a departure from the rectangular mode of 
surveying and subdividing the public lands in California 
would promote the public interests, he may direct such 
change to be made in the mode of surveying and desig¬ 
nating such lands as he deems proper, with reference to 
the existence of mountains, mineral deposits, and the ad¬ 
vantages derived from timber and water privileges; but 
such lands shall not be surveyed into less than one hun- 



ORIGINAL SURVEYS. 185 

dred and sixty acres or subdivided into less than forty 
acres. 

10 Stat. 245: R. S. 2410. Cir. G. L. O., June 26, 1880. 

Sec. 106. The public surveys shall extend over all min¬ 
eral lands, and all subdividing of surveyed lands into lots 
less than one hundred and sixty acres may be done by 
county and local surveyors at the expense of claimants; 
but nothing contained in this section shall require the 
survey of waste or useless lands. 

10 stat. 15, 21; 16 id. 218; R. S. 2406. 

Sec. 107. The printed manual of instructions relating 
to tne pujohc surveys, prepared at the General Land Office, 
and bearing date June thirtieth, eighteen hundred and 
ninety-four, the instructions of the Commissioner of 
the General Land Office, and the special instructions 
of the surveyor-general, when not in conflict with such 
printed manual or the instructions of the Commissioner, 
shall be taken and deemed to be a part of every contract 
for surveying the public lands. 

12 Stat. 409; R. S. 2399. Cir. G. L. O., June 26, 1880. 

Sec. 108. Legal subdivisions of forty acres of placer 
lands may be subdivided into ten-acre lots. 

16 Stat. 213; R. S. 2330. 

Sec. 2320. Mining claims upon veins or lodes of quartz 
or other rock in place bearing gold, silver, cinnabar, lead, 
tin, copper, or other valuable deposits, heretofore located, 
shall be governed as to length along the vein or lode by 
the customs, regulations, and laws in force at the date of 
their location. A mining-claim located after the tenth 
day of May, eighteen hundred and seventy-two, whether 
located by one or more persons, may equal, but shall not 
exceed, one thousand five hundred feet in length along 
the vein or lode; but no location of a mining-claim shall 
be made until the discovery of the vein or lode within 
the limits of the claim located. ]STo claim shall extend 
more than three hundred feet on each side of the middle 


186 


A MANUAL OF LAND SURVEYING. 


of the vein at the surface, nor shall any claim be limited 
by any mining regulation to less than twenty-five feet on 
each side of the middle of the vein at the surface, except 
where adverse rights existing on the tenth day of May, 
eighteen hundred and seventy-two, render such limita¬ 
tion necessary. The end-lines of each claim shall be 
parallel to each other. 

10 May, 1872, c. 152, S. 2, V. 17, p. 91. 

Sec. 2322. The locators of all mining locations hereto¬ 
fore made or which shall hereafter be made, on any min¬ 
eral vein, lode, or ledge, situated on the public domain, 
their heirs and assigns, where no adverse claim exists on 
the tenth day of May, eighteen hundred and seventy-two, 
so long as they comply with the laws of the United 
States, and with State, Territorial and local regulations 
not in conflict with the laws of the United States govern¬ 
ing their possessory title, shall have the exclusive right 
of possession and enjoyment of all the surface included 
within the lines of their locations, and of all veins, lodes, 
and ledges throughout their entire depth, the top or apex 
of which lies inside of such surface-lines extended down¬ 
ward vertically, although such veins, lodes, or ledges may 
so far depart from a perpendicular in their course down¬ 
ward as to extend outside the vertical side-lines of such 
surface locations. But their right of possession to such 
outside parts of such veins or ledges shall be confined to 
such portions thereof as lie between vertical planes 
drawn downward as above described, through the end¬ 
lines of their locations, so continued in their own direc¬ 
tion that such planes will intersect such exterior parts of 
such veins or ledges. And nothing in this section shall 
authorize the locator or possessor of a vein or lode which 
extends in its downward course beyond the vertical lines 
of his claim to enter upon the surface of a claim owned 
or possessed by another. 

10 May, 1872, C. 152, S. 3, V. 17, p. 91. 

Sec. 2323. Where a tunnel is run for the development 
of a vein or lode, or for the discovery of mines, the own- 



ORIGINAL SURVEYS. 


1ST 


ers of such tunnel shall have the right of possession of 
all veins or lodes within three thousand feet from the 
face of such tunnel on the line thereof, not previously 
known to exist, discovered in such tunnel, to the same 
extent as if discovered from the surface; and locations 
on the line of such tunnel of veins or lodes not appearing 
on the surface, made by other parties after the commence¬ 
ment of the tunnel, and while the same is being prose¬ 
cuted with reasonable diligence, shall be invalid; but 
failure to prosecute the work on the tunnel for six 
months shall be considered as an abandonment of the 
right to all undiscovered veins on the line of such tunnel. 

10 May, 1872, C. 152, s. 4, V. 17, p. 92. 

Sec. 2324. The miners of each mining-district may 
make regulations not in conflict with the laws of the 
United States, or with the laws of the State or Territory 
in which the district is situated, governing the location, 
manner of recording, amount of work necessary to hold 
possession of a mining-claim, subject to the following 
requirements: The location must be distinctly marked on 
the ground so that its boundaries can be readily traced. 
All records of mining-claims hereafter made shall con¬ 
tain the name or names of the locators, the date of the 
location, and such a description of the claim or claims 
located by reference to some natural object or permanent 
monument as will identify the claim. 

10 May, 1872, C. 152, S. 5, V. 17, p. 92. 

Sec. 109. The surveyor-general of the United States 
may appoint in each land district containing mineral 
lands as many competent surveyors as shall apply for ap¬ 
pointment to survey mining claims. The expenses of the 
survey of vein or lode claims, and the survey and sub¬ 
division of placer claims into smaller quantities than one 
hundred and sixty acres, shall be paid by the applicants, 
and they shall be at liberty to obtain the same at the 
most reasonable rates, and they shall also be at liberty to 
employ any United States deputy surveyor to make the 


188 


A MANUAL OF LAND SURVEYING. 


survey. The Commissioner of the General Land Office 
shall have power to establish the maximum charges for 
such surveys; and to the end that he may be fully in¬ 
formed on the subject, each applicant shall file with the 
register a sworn statement of all charges and fees paid 
by such applicant for surveys, which statement shall be 
transmitted to the Commissioner of the General Land 
Office. 

17 Stat. 95; 19 id. 52; K. S. 2334. Decision Coin. G. L. ()., April 20, 
1877. 

Sec. 110. The surveyor-general of the United States 
shall prepare or cause to be prepared a plat and field-notes 
of all mining surveys made by authority of law, which 
shall show accurately the boundaries of such claims; and, 
when warranted by the facts, he shall give to the claim¬ 
ant his certificate that five hundred dollars’ worth of 
labor has been expended or improvements made upon the 
claim by the claimant or his grantors, and that the plat 
is correct, with such further description by such refer¬ 
ence to natural objects or permanent monuments as shall 
identify the claim, and furnish an accurate description, 
to be incorporated in the patent. 

17 Stat. 92 11. S. 2325 

Sec. 111. Contracts for the survey of the public lands 
shall not become binding upon the United States until 
approved by the Commissioner of the General Land 
Office, except in such cases as the Commissioner may 
otherwise specially order. 

12 Stat. 409; R. S, 2398. Maguire v. Tyler, 1 Black, 201; Barks v. Ross, 
11 How. 362*; Spencer v. Lapsley, 20 id 204. Reed v. Conway, 26 
Mo. 13. Decision Sec. Int., Feb. 27, 1878. 

Sec, 112. The Commissioner of the General Land Office 
has power, and it shall be his duty, to fix the prices per 
mile for public surveys, which shall in no case exceed the 
maximum established by law; and, under instructions to 
be prepared by the Commissioner, an accurate account 
shall be kept by each surveyor-general of the cost of sur- 


ORIGINAL SURVEYS. 


189 


veying and platting private land claims, to be reported to 
the General Land Office, with the map of such claim; 
and patents shall not issue for any such private claim, 
nor shall any copy of such survey be furnished, until the 
cost of survey and platting has been paid into the Treas¬ 
ury by the claimant or other party; and before any land 
granted to any railroad company by the United States 
shall be conveyed to such company or any persons entitled 
thereto, under any of the acts incorporating or relating to 
said company, unless such company is exempted by law 
from the payment of such cost, there shall first be paid 
into the Treasury of the United States the cost of sur¬ 
veying, selecting, and conveying the same by the said 
company or persons in interest. 

12 Stat. 409; 18 id. 3S4; 19 id. 122; R. S. 2400, Railway Co. v. Prescott, 
16 Wall. 603; Railway Co. v. McShane, 22 id. 444; Hannewell v. 
Cass Co., 22 id. 464; Colorado Co. v. Commissioners, 5 Otto, 259. 
Decisions Sec. Int., Dec. 17, 1874; Feb. 27, 1878; Feb. 20, 1879; 
March 5, .1879; April 2, 1879. Decisions Com. G. L. O., April 18, 
1867; August 18, 1867; Feb. 17, 1869; March 26, 1870. Cir. G. L. O., 
June 26,1880. 

Sec. 113, The Commissioner of the General Land Office 
may authorize, in his discretion, public lands in Oregon 
densely covered with forests or thick undergrowth, to be 
surveyed at augmented rates, not exceeding eighteen dol¬ 
lars per mile for standard parallels, fifteen dollars for 
townships, and twelve dollars for section lines; and 
under like conditions he may allow augmented rates in 
California, and in Washington Territory, not exceeding 
eighteen dollars per linear mile for standard parallels, 
sixteen dollars for township, and fourteen dollars for 
section lines. 

16 Stat. 304, 305 ; 17 id. 358 ; R. S. 2404, 2405. Decision Sec. Int., June 
16, 1879. Cir. G. L. O., June 26, 1880. 

Sec. 114. Whenever the public surveys, or any portion 
of them, in the States of Oregon and California, are so 
required to be made as to render it expedient to make, 
compensation for the surveying thereof by the day instead 


190 


A MANUAL OF LAND SURVEYING. 


of by the mile, it shall be lawful for the Commissioner of 
the General Land Office, under the direction of the Secre¬ 
tary of the Interior, to make such fair and reasonable 
allowance, as, in his judgment, may be necessary to insure 
the accurate and faithful execution of the work. 

lOStat. 247; R. S. 2411. Decision Sec. Int., June 16, 1879. Cir. G. L. 
O., June 26,1880. 

Sec. 118. Each surveyor-general, when thereunto duly 
authorized by law, shall cause all confirmed private land 
claims within his district to be accurately surveyed, and 
shall transmit plats and field-notes thereof to the Com¬ 
missioner of the General Land Office for his approval. 
When publication of such surveys is authorized by law, 
the proof thereof, together with any objections properly 
filed and all evidence submitted either in support of or in 
opposition to the approval of any such survey, shall also 
be transmitted to said Commissioner. 

2 Stat. 326, 352; 3 id, 325 ; 5 id. 740; 9 id. 242, 633 ; 10 id. 244, 308, 599; 

11 id. 294; 12 id. 172, 209, 369,409; 13 id. 332,344; 14 id. 218; 16 id. 
64, 304; 18 id. 305; 19 id. 121, 202: R. S. 2447. Bissell v. Penrose, 8 
How. 317; Villalobus v. U. S., 10 id. 541; Ledoux v. Black, 18 id. 
473; U. S. v. Fossat, 20 id. 413; Brown v. Huger, 21 id. 305; U. S. v. 
Fossat, 21 id, 445, Castro v. Hendricks, 23 id. 438; Ballance v. For¬ 
syth, 24 id. 183; U. S. v. Sepulveda, 1 Wall. 104; U. S. v. Halleck, 
l id. 439; U. S. v. Vallejo, 1 id. 658; Sutter's case 2 id. 562; Fossat 
case, 2 id, 649; Higueras v. U. S , 5 id. 827; Alviso v. U. S., 8 id. 337. 

12 Op. Att. Gen. 116, 250; 14 id, 74, 601. U. S, v. Garcia, 1 Saw. C.C. 
383; Russell v. Henshaw, 1 id, 553; Leroy v. Jamison, 3 id. 369; 
U. S. v. Flint, 4 id. 42. Dent v. Sergerson, 29 Mo. 480; Fowler v. 
Duvall, 11 La. Ann. 561; Waterman v. Smith, 13 Cal. 373; Moore v 
Wilkerson, 13 id. 478; Merrit v. Judd, 14 id. 60; Mott v. Smith, lOid. 
534; Johnson v. Van Dyke, 20 id. 225; McGarraghan v. Maxwell, 
27 id. 75 ; Seale v. Ford, 29 id. 104. Cir. G. L. O., June 26, 1880. 

Sec. 120. Every person who in any manner, by threat 
or force, interrupts, hinders, or prevents the surveying of 
the public lands, or of any private land claim which has 
been or may be confirmed by the United States, by the 
persons authorized to survey the same, in conformity 
with the instructions of the Commissioner of the General 



ORIGINAL SURVEYS. 


191 


Land Office, shall be fined not less than fifty dollars nor 
more than three thousand dollars, and be imprisoned not 
less than one nor more than three years. 

4 stat. 417; R, S. 2412. 

Sec. 121. Whenever the President is satisfied that forci¬ 
ble opposition has been offered, or is likely to be offered, 
to any surveyor or deputy surveyor in the discharge of 
his duties in surveying the public lands, it may be lawful 
for the President to order the marshal of the State or 
district, by himself or deputy, to attend such surveyor 
or deputy surveyor with sufficient force to protect such 
officer in the execution of his duty, and to remove force 
should any be offered. 

4 stat. 417 , R. S. 2413. 

Sec. 122. The President is authorized to appoint sur¬ 
veyors of public lands, who shall explore such vacant 
and unappropriated lands of the United States as produce 
the live-oak and red-cedar timbers, and shall select such 
tracts or portions thereof, where the principal growth is 
of either of such timbers, as in the judgment of the Sec¬ 
retary of the Navy may be necessary to furnish for the 
Navy a sufficient supply of the same. Such surveyors 
shall report to the President the tracts by them selected, 
with the boundaries ascertained and accurately desig¬ 
nated by actual survey or water-courses. 

3 Stat. 347; R. S. 2459. IT. S. v. Briggs, 9 How. 351. 

Sec. 123. The director of the geological survey shall, 
under the Interior Department, have the direction of the 
geological survey and the classification of the public 
lands and examination of the geological structure, min¬ 
eral resources, and products of the national domain. 

20 Stat. 394. 

8. Manner of Field Work and Changes that 
have been Made.— In accordance with these laws, in¬ 
structions have been issued from time to time, by the 


192 


A MANUAL OF LAND SURVEYING. 


Commissioners of the General Land Office, directing the 
manner in which the field work should be performed. 

In the earlier surveys under the act of 1796 (Sec. 2395 
R. S. See p. 180, Sec. 99, Third,) the township was sub¬ 
divided by parallel lines two miles apart. The mile posts 
were planted on these lines, but no half mile (or quarter- 
section) corners set. 

The act of 1800 provided that the townships west of 
the Muskingum River should be subdivided into half 
sections of 320 acres each, as near as may be, by parallel 
lines run through them from east to west and from north 
to south at distances of a mile apart. Half-mile posts 
were to be set on the east and west lines, but not .on the 
lines running north and south. 

The act of 1805 (Sec. 2396 R. S. P. 181, Sec. 100) covers 
in its provisions the two classes of surveys above noted, 
as well as the principles governing all subsequent surveys 
of the public lands. 

Since that time, few changes have been made in the 
manner of carrying on the surveys. 

The principal change has been in the manner of closing 
the subdivision lines on the exterior line of the township. 

In some of the earlier surveys, three sets of corners 
were marked in the range lines. The first set was marked 
when the range line was run, and were not really corners 
of the subdivisions. 

The other two sets were marked at the points where 
the subdivision lines of. the townships, both east and 
west, intersected the range line—those lines not being 
required to close on the corners previously set on the 
range line. 

Later the surveyors were required to close their subdi¬ 
vision lines upon the corners previously set on the east 
line of the township, but not on the north or west. 
Double corners were thus produced on all the exterior 
lines of the township. 


ORIGINAL SURVEYS. 


193 


Most of the surveys before 1846 were made under this 
system, which is thus laid down in the Instructions of 
1815: 

Each side of a section must be made one mile in 
measure by the chain, and quarter-section corners are to 
be established at every half mile, except when in the 
closing 1 of a section if the measure of the closing side 
should vary from 80 chains or one mile, you are in that 
case to place the quarter-section corners equidistant, or 
at an average distance from the corners of the section; 
but in running out the sectional lines on the west or north 
side of the township, you will establish your quarter- 
section posts or corners at the distance of half a mile 
from the last corner, and leave the remaining excess or 
defect on the west or north tier of quarter-sections, which 
balance or remainder you will carefully measure and put 
down in your field-notes in order to calculate the remain¬ 
ing or fractional quarter-section on the north and west 
side of the township: also in running to the western or 
northern boundary, unless your sectional lines fall in with 
the posts established there for the corners of sections in 
the adjacent townships, you must set post and mark 
bearing trees at the points of intersection of your lines 
with the town boundaries, and take the distance of your 
corners from the corners of the sections of the adjacent 
townships, and note that and the side on which it varies 
in chains or links, or both. 

The sections must be made to close by running a ran¬ 
dom line from one corner to another, except on the north 
and west ranges of sections, and the true line between 
them is to be established by means of offsets.” 

Under the present system, which has been in use in 
some parts of the country since 1846, the section lines 
are required to close on the corners previously set on the 
north and west boundaries, the same as on the east, thus 
doing away with the system of double section corners. 


14 


194 


A MANUAL OF LAND SURVEYING. 


The practice in the several surveying districts in the 
United States does not seem to have been uniform at any 
time previous to 1860, and perhaps not always since that 
date. For instance, in the Instructions of the Commis¬ 
sioner of the General Laiid Office to surveyors-general, 
dated Feb. 22, 1855, which is stated to be a revision of 
the manual of surveying instructions prepared for Or¬ 
egon in 1851, it is expressly ordered that “double corners 
are to be nowhere except on the base and standard lines;” 
while in the instructions to deputy surveyors of the 
United States for the district of Illinois and Missouri, 
published in 1856, P. 9, the deputy surveyors were 
directed to plant their closing corners at the intersec¬ 
tion of their lines with the north and west boundary and 
return their direction and distance from the corners of 
the corresponding sections on the north and west of 
these boundaries,” the surveyor-general of that district 
thus giving different instructions from those of the 
Commissioner of the General Land Office. 

9. Fractional Areas. —It has been a puzzle to 
many surveyors to know how the area of the fractional 
quarter-sections adjoining the north and west boundaries 
of the township were calculated. It has been just as 
much of a puzzle to the surveyors-general and Commis¬ 
sioners of the General Land Office. 

Edward Tiffin, surveyor-general of the Northwest 
Territory, in 1815 issued Instructions how to do it, which 
instructions were made applicable to the surveys in Ohio, 
Michigan, Arkansas and Missouri. Under these instruc¬ 
tions, the calculations of the areas of these fractions 
were to be made on the assumption that the quarter- 
posts on the township and range lines were common to 
the sections on both sides of these lines, thus making 
the lengths of the fractions more or less unequal where 
there were double section corners. This plan does not 
seem to have been in force long, or to have been very 
generally followed. Another plan quite extensively 
adopted was to make the calculations on the theory that 
all the north and south quarter-lines of these fractional 
sections were to be parallel with the east line of the sec- 




ORIGINAL SURVEYS. 


195 


tions, and all east and west quarter-lines parallel with the 
south line of the sections. Neither plan was in harmony 
with the law of 1805, which required “ the corners of half 
and quarter sections not marked on the surveys to be 
placed as nearly as possible equidistant from those two 
corners which stand on the same line.” 

The plan under which most if not all the fractional 
areas of Michigan were calculated was on the theory that 
the quarter-posts on the township and range lines were to 
be placed midway between their respective section corners. 

Previous to 1828, the deputy surveyors were required to 
return with their field notes plats of all the townships 
which they surveyed, and to calculate the area of the 
fractions. These plats were rudely constructed, and in 
many cases the areas put down on them were erroneous. 
If this was found out before the land was sold, the areas 
were re calculated in the surveyor-general’s office. In 
making the calculations of the areas of the fractions 
along the township and range lines, some of the deputies 
considered the quarter-section corners along those lines 
as common to the sections on both sides, some adopted 
the second method described above, while the areas of 
many of the fractions appear to have been put down 
without any calculation whatever. 

In the U. S. Surveying Instructions of June 30, 1894, 
the following rules are given : — 

In the north tier of Sections the fractional lots along 
the boundary are numbered 1 to 4 from east to west. 
In the west tier they are numbered from north to 
south. In Section 6 they are numbered from 1 to 7 
from the N. E. corner of the Section along the boundary 
to the S. W. corner. 

1. In regular townships, the tracts of land in each sec¬ 
tion adjoining the north and west boundaries of such 
townships, in excess of the regularly subdivided 480 
acres (except in section 6), will, in general, be in the 
form of trapezoids, 80.00 chains in length by about 20 
chains in width. 

' On the plats of such townships, each of said tracts 
will be divided into four lots, by drawing broken lines 


196 


A MANUAL OF LAND SURVEYING. 


at intervals of 20.00 chains, parallel to the ends of the 
tracts, which will be regarded as parallel to each other. 

With the exception of section 6, the south boundaries 
of sections of the north tier, when within prescribed 
limits, will be called 80.00 chains. 

When the above-named conditions obtain, the areas 
of the lots in any one tract (except in section 6) may be 
determined, as follows: — 

Divide the difference between the widths of the ends 
of the tract by 4; if 3 remains, increase the hundredth 
figure of the quotient by a unit; in all other cases disre¬ 
gard the fraction; call the quotient thus obtained, “d ;” 
then, taking the end widths of the tract in chains and 
decimals of a chain, the areas of the lots, in acres, will be: — 

Of the smallest lot: twice the width of the lesser end, 
plus “ d; ” 

Of the largest lot: twice tbe width of the greater end, 
minus “ d; ” 

Of the smaller middle lot: sum of the widths of the 
ends, minus “d; ” 

Of the larger middle lot: sum of the widths of the ends, 
plus “d.” 

A check on the computation may be had by multiply¬ 
ing the sum of the widths of the ends of the tract by 4; 
the product should agree exactly with the total area of 
the four lots. 

The proper application of the above rules will always 
give areas correct to the nearest hundredth of an acre; and, 
as the use of fractions is entirely avoided, the method 
is recommended for its simplicity and accuracy. 

Example 1. 

The i difference of latitudinal boundaries is 0.031 
chains: consequently, “d ” is .04 chains; then, 

18.35X 2 +.04= 36.74 acres, the area of lot 1; 

18.50X 2 —.04= 36.96 acres, the area of lot 4: 

18.50+18.35 —.04= 36.81 acres, the area ol' lot 2: 

18.50+18.35 +.04 = 36.89 acres, the area of lot 3; 

Check: [18.35+18 50] x 4 = 147.40 acres, the area of the four lots. 

The arithmetical operations are here written in de¬ 
tail, for the purpose of illustration; but the practical 
computer will perform all the work mentally. 



ORIGINAL SURVEYS. 


197 


2. Section 6. The areas of lots 5, 6, and 7 may be ob¬ 
tained by the foregoing rules in all cases, except when 
the township closes on a base line or standard par¬ 
allel; also, the area of lot 4, provided both meridional 
boundaries are 80.00 chains in length; when the last 
condition obtains, the areas of lots 1, 2, and 3 will be 
equal, and each will contain 40.00 acres. 

In any case where the west boundary of sec. 6, is 80.00 
chains, and the east boundary either greater or less than 
mod chains, the areas of lots 1, 2, 3. and 4 will be com¬ 
puted as follows: — 

Determine the difference, “q,” between the east 
boundaries of lots 1 and 4 by the following propor¬ 
tion:— 

N. bdy. sec. 6.: diff. of meridional bdrs. sec. 6. ::60chs. : 
q; then will E. bdy. lot 4—E. bdy. lot l±q; in which, 
“q ” w ill be added when the east boundary of sec. 6 is 
less than 80.00 chains; but subtracted wdien said east bound¬ 
ary is greater than 80.00 chains. 

Now take one third, of “q,”and add it to the shorter 
east boundary of lots 1 or 4, as conditions may require, 
and thereby determine the length of one of the meridi¬ 
onal boundaries of lot 2; to which again add “one 
third of q,” and thus obtain the length of the opposite 
side of lot 2. The areas of lots 1, 2, and 3, in acres, will 
be found by taking the sum of their respective meridi¬ 
onal boundaries, expressed in chains and decimals of a chain. 

The area of lot 4 may be had by multiplying its mean 
width by its mean length. 

Finally, to test the entire work, multiply the sum of 
the latitudinal boundaries by 4, and to the product add 
the area of the small triangle C A B, if the east boun¬ 
dary is greater than 80.00 chains; but subtract the area of 
said small triangle if the east boundary is less than 80.00 
chains. These operations, correctly performed, will 
give the true area of the section, which should agree exactly 
with the total area of its legal subdivisions, obtained as 
directed in the preceding paragraphs. 

Example 2. 

Compute areas of lots 5, 6, and 7 of sec. 6, as directed 


198 


A MANUAL OF LAND SURVEYING. 


in paragraph 1, and illustrated by the example; then 
write: — 

chs. chs. chs. chs. chs. 

77.75 : 0.05 :: 60.00 : 0.0386 = q; H q=0.0129 

chs. chs. chs. 

20.0500—0.0386 = 20.01, the E. bdy. of lot 4; 

20.0114+0.0129=20.02, the E. bdy. of lot 3; 

20.0243+0.0129=20.04, the E. bdy. of lot 2. 

Then, for the areas of lots 1, 2, 3, and 4, we have: — 

chs. chs. acres. 

20.05+20.04.= 40.09, the area of lot 1; 

20.04+20.02. = 40.06, the area of lot 2; 

20.02+20.01 .= 40.03, the area of lot 3; 

20,00+20.01 17.75+17.78 = 35 M the „ ea of lot 4 

2 2 

Also [17.78+17.87] X3 =106.95. the area of lots 5, 6 , and 7. 

Area of regular subdivisions=360.00 

Total.=622.67, the area of Sec. 6 . 

clis. chs. 

Check: [77.87+77.75]X4=622.48 

77.75 X 0.025 = 0.19, the area of triangle C A B. 

Total.=622.67, which agrees with the area of section 6 , 

before determined. 

3. The area in acres of a tract 40.00 chains long, ad¬ 
joining north or west township boundaries (except in 
N. W. 4 sec. 6), is equal to the sum of its parallel bound¬ 
aries (expressed in chains and decimals thereof) multi¬ 
plied by 2; (e. g.) the area of lots 6 and 7, is [17.87+17.81] 
X 2=71.30 acres. 

The area in acres of a tract 00.00 chains long, situated 
as above described (excluding lot 4, of sec. 6), may be 
* found by multiplying the sum of its parallel boundaries 
(expressed in chains and decimals of a chain) by 3; (e. g.) 
Fig. 6; south boundary lot 4=17.78 chs.; area of lots 5, 6, 
and 7 is [17.78+17.87]x3=106.95 acres. (See example 2.) 

The area in acres of quarter sections adjoining north 
and west township boundaries (excluding N. W. 4 sec. 
6), may be obtained by multiplying the sum of their 
parallel boundaries (taken in chains and decimals of a 
chain), by 2; (e. g.) the area of S. W. 4 sec, 6 (Fig. 0), is 
[37.87+37.81]x2=151.30 acres. 

The area in acres of any section along the north and 
west boundaries of regular townships (except sec. 6) may 


i 













ORIGINAL SURVEYS. 


199 


be had by multiplying the sum of its parallel boundaries 
(expressed in chains and decimals of a chain) by 4; (e. g.) 
the area of sec. 1 (Plate IV) is [80.00-f79.77]x4=639.08 
acres. 

The area in acres of a theoretical township may be ob¬ 
tained by multiplying the sum of its latitudinal hound- 
ernes (expressed in chains and decimals of a chain) by 24 
(e. g.) the area of a township is [480.00-f-479.34]x24=23, 
024.16 acres. 

lO. Instructions of 1894.—The U. S. Manual of 
Surveying Instructions for 1894, is a large volume of 236 
pages and contains minute instructions in regard to all 
the operations of the survey of the public lands and 
private land claims. It is furnished to Deputy U. S. 
Surveyors and may be had by others who apply for it to 
the Commissioner of the General Land Office at Wash¬ 
ington. The following extracts are made from it: — 

SYSTEM OF RECTANGULAR SURVEYING. 

1. Existing law requires that in general the public 
lands of the United States “ shall be divided by north 
and south lines run according to the true meridian, and 
by others crossing them at right angles so as to form 
townships six miles square,’’and that the corners of the 
townships thus surveyed “must be marked with pro¬ 
gressive numbers from the beginning.” 

Also, that the townships shall be subdivided into 
thirty-six sections, each of which shall contain six hun¬ 
dred and forty acres, as nearly as may be, by a system of 
two sets of parallel lines, one governed by true meridi¬ 
ans and the other by parallels of latitude, the latter in¬ 
tersecting the former at right angles, at intervals of a 
mile. 

2. In the execution of the public surveys under exist¬ 
ing law, it is apparent that the requirements that the 
lines of survey shall conform to true meridians, and that 
the townships shall be 6 miles square, taken together, 
involve a mathematical impossibility due to the con- 
vergency of the meridians. 




200 


A MANUAL OF LAND SURVEYING. 


Therefore, to conform the meridianal township lines 
to the true meridians produces townships of a trape¬ 
zoidal form which do not contain the precise area of 
23,040 acres required by law, and which discrepancy in¬ 
creases with the increase in the convergency of the 
meridians, as the surveys attain the higher latitudes. 

In view of these facts, and under the provisions of 
section 2 of the act of May 18, 1796, that sections of a 
mile square shall contain 640 acres, as nearly as may be, 
and also under those of section 3 of the act of May 10, 
1800, that “in all cases where the exterior lines of the 
townships, thus to be subdivided into sections and half 
sections, shall exceed, or shall not extend 6 miles, the 
excess or deficiency shall be specially noted, and added 
to or deducted from the western or northern ranges of 
sections or half sections in such township, according as 
the error may be in running lines from east to west, or 
from south to north ; the sections and half sections 
bounded on the northern and western lines of such 
townships shall be sold as containing only the quantity 
expressed in the returns and plats, respectively, and all 
others as containing the complete legal quantity,” the 
public lands of the United States shall be surveyed un¬ 
der the methods of the system of rectangular surveying, 
which harmonizes the incompatibilities of the require¬ 
ments of law and practice, as follows: — 

First, The establishment of a principal meridian con¬ 
forming to the true meridian, and, at right angles to it, 
a base line conforming to a parallel of latitude. 

Second. The establishment of standard parallels con¬ 
forming to parallels of latitude, initiated from the 
principal meridian at intervals of 24 miles and extended 
east and west of the same. 

Third. The establishment of guide meridians con¬ 
forming to true meridians, initiated upon the base line 
and successive standard parallels at intervals of 24 miles, 
resulting in tracts of land 24 miles square, as nearly as 
may be, which shall be subsequently divided into tracts 
of land 6 miles square by two sets of lines, one conform¬ 
ing to true meridians, crossed by others conforming to 


ORIGINAL SURVEYS. 


201 


parallels of latitude at intervals of 6 miles, containing 
23,040 acres, as nearly as may be, and designated townships. 

Such townships shall be subdivided into thirty-six 
tracts, called sections, each of which shall contain 640 
acres, as nearly as may be, by two sets of parallel lines, 
one set parallel to a true meridian and the other conforming 
to parallels of latitude, mutually intersecting at intervals 
of 1 mile and at right angles, as nearly as may be. 

Any series of contiguous townships situated north 
i and south of each other constitutes a range, while such 
a series situated in an east and west direction consti¬ 
tutes a tier. 

The section lines are surveyed from south to north and 
from east to west, in order to throw the excess or defi¬ 
ciency in measurement on the north and west sides of 
the township, as required by law. In case where a town¬ 
ship has been partially surveyed, and it is necessary to 
complete the survey of the same, or where the character 
of the land is such that only the north or west portions 
of the township can be surveyed, this rule cannot be 
strictly adhered to, but, in such cases, it will be de¬ 
parted from only so far as is absolutely necessary. It 
will also be necessary to depart from this rule where 
surveys close upon State or Territorial boundaries, or 
upon surveys extending from different meridians. 

3. The tiers of townships will be numbered, to the 
north or south, commencing with No. 1, at the base 
line; and the ranges of the townships, to the east or 
west, beginning with No. 1, at the principal meridian 
of the system. 

4. The thirty-six sections into which a township is 
subdivided are numbered, commencing with number 
one at the northeast angle of the township, and proceed¬ 
ing west to number six, and thence proceeding east to 
number twelve, and so on, alternately, to number thir¬ 
ty-six in the southeast angle. In all cases of surveys of 
fractional townships, the sections will bear the same 
numbers they would have if the township was full. 

5. Standard parallels shall be established at intervals 
of every 24 miles, north and south of the base line, and 


202 


A MANUAL OF LAND SURVEYING. 

guide meridians at intervals of every 24 miles, east and 
west of the principal meridian; thus confining the er¬ 
rors resulting from convergence of meridians and inac¬ 
curacies in measurement within comparatively small 
areas. 

Instruments.— 6. The surveys of the public lands 
of the United States, embracing the establishment of 
base lines, principal meridians, standard parallels, mean¬ 
der lines, and the subdivisions of townships, will be 
made with instruments provided with the accessories 
necessary to determine a direction with reference to the 
true meridian, independently of the magnetic needle. 

Burt’s improved solar compass, or a transit of ap¬ 
proved construction, with or without solar attachment, 
will be used in all cases. When a transit without solar 
attachment is employed, Polaris observations and the re¬ 
tracements necessary to execute the work in accordance 
with existing law and the requirements of these instruc¬ 
tions will be insisted upon. 

7. Deputies using instruments with solar apparatus 
will be required to make observations on t he star Polaris 
at the beginning of every survey, and, whenever necessary, 
to test the accuracy of the solar apparatus. 

The observations required to test the adjustments of 
the solar apparatus will be made at the corner where 
the survey begi ns, or at the camp of the deputy surveyor 
nearest said corner; and in all cases the deputy will 
fully state in the held notes the exact location of the 
observing station. 

Deputy surveyors will examine the adjustments of 
their instruments, and take the latitude daily, weather per¬ 
mitting, while running all lines of the public surveys. They 
will make complete records in their field notes, under 
proper dates, of the making of all observations in com¬ 
pliance with these instructions, showing the character 
and condition of the instrument in use, and the precis¬ 
ion attained in the survey, by comparing the direction 
of the line run with the meridian determined by obser¬ 
vation. 


ORIGINAL SURVEYS. 


203 


On every survey executed with solar instruments, the 
deputy will, at least once on each working day , record in liis 
( field notes the proper reading of the latitude arc; the 
declination of the sun, corrected for refraction, set off 
on the declination arc; and note the correct local mean 
time of his observation, which, for the record, will be 
taken at least two hours f rom apparent noon. 

8. The construction and adjustments of all surveying 
instruments used in surveying the public lands of the 
United States will be tested at least once a year, and 
oftener, if necessary, on the true meridian, established 
under the direction of the surveyor general of the dis¬ 
trict; and if found defective, the instruments shall un¬ 
dergo such repairs or modifications as may be found 
necessary to secure the closest possible approximation 
to accuracy and uniformity in all field work controlled 
by such instruments. 

A record will be made of such examinations, showing 
the number and character of the instrument, name of 
the maker, the quantity of instrumental error discov¬ 
ered by comparison, in either solar or magnetic appa¬ 
ratus, or both, and means taken to correct the same. 
The surveyor general will allow no surveys to be made 
until the instruments to be used therefor have been 
approved by him. 

9. The township and subdivision lines will usually be 
measured by a two-pole chain 33 feet in length, consist¬ 
ing of 50 links, each link being seven and ninety-two 
hundredths inches long. On uniform and level ground, 
however, the four-pole chain may be used. The meas¬ 
urements will, however, always be expressed in terms of 
the four-pole chain of 100 links. The deputy surveyor 
shall provide himself with a measure of the standard 
chain kept at the office of the surveyor general, to be 
used by him as a field standard. The chain in use will 
be compared and adjusted with this field standard each 
working day, and such field standard will be returned 
to the surveyor general’s office for examination when 
the work is completed. 






204 


A MANUAL OR LAND SURVEYING. 


Deputy surveyors will use eleven tally pins made of 
steel, not exceeding 14 inches in length, weighty enough 
toward the point to make them drop perpendicularly, 
and having a ring at the top, in which will be fixed a 
piece of red cloth, or something else of conspicuous 
color, to make them readily seen wdien stuck in the 
ground. 

Process of Chaining. — In measuring lines with 

a two-pole chain, five chains are called a “tally; ” and in 
measuring lines with a four-pole chain, ten chains are 
called a “tally,” because at that distance the last of the 
ten tally pins with which the forward chainman sets 
out will have been stuck. He then cries “ tally,” which 
cry is repeated by the other chainman, and each regis¬ 
ters the distance by slipping a thimble, button, or ring 
of leather, or something of the kind, on a belt worn for 
that purpose, or by some other convenient method. 
The hind chainman then comes up, and having counted 
in the presence of his fellow, the tally pins which he has 
taken up, so that both may be assured that none of the 
pins have been lost, he then takes the forward end of 
the chain, and proceeds to set the pins. Thus the 
chainmen alternately change places, each setting the 
pins that he has taken up, so that one is forward in all 
the odd, and the other in all the even tallies. Such pro¬ 
cedure, it is believed, tends to insure accuracy in meas¬ 
urement, facilitates the recollection of the distances 
to objects on the line, and renders a mistally almost 
impossible. 

Leveling the Chain and Plumbing the Pins. 

— 1. The length of every surveyed line will be ascer¬ 
tained by precise horizontal measurement, as nearly ap¬ 
proximating to an airline as is possible in practice on 
the earth’s surface. This all-important object can only 
be attained by a rigid adherence to the three following 
observances: — 

First. Ever keeping the chain drawn to its utmost de¬ 
gree of tension on even ground. 

Second. On uneven ground, keeping the chain not only 
stretched as aforesaid, but leveled. And when ascending 


ORIGINAL SURVEYS. 


205 


and descending steep ground, hills, or mountains, the 
chain will have to be shortened enough to accurately ob¬ 
tain the true horizontal measure. 

Third. The careful plumbing of the tally pins, so as 
to attain precisely the spot where they should be stuck. 
The more uneven the surface, the greater the caution 
needed to set the pins. 

Marking Lines.— 1. All lines on which are to be 
established the legal corner boundaries will be marked 
after this method, viz.: Those trees which may be inter¬ 
sected by the line, will have two chops or notches cut 
on the sides facing the line, without any other marks 
whatever. These are called “sight trees” or “line trees.” 
A sufficient number of other trees standing within 50 
links of the line, on either side of it, will be blazed on 
two sides diagonally or quartering toward the line, in 
order to render the line conspicuous, and readily to be 
traced; the blazes to be opposite each other, coinciding 
in direction with the line where the trees stand very 
near it, and to approach nearer each other toward the 
line, the farther the line passes from the blazed trees. 
Due care will ever be taken to have the lines so well 
marked as to be readily followed, and to cut the blazes 
deep enough to leave recognizable scars as long as the 
trees stand. 

Where trees 2 inches or more in diameter are found, 
the required blazes will not be omitted. 

Bushes on or near the line should be bent at right 
angles therewith, and receive a blow of the ax at about 
the usual height of blazes from the ground, sufficient to 
leave them in a bent position, but not to prevent their 
growth. 

2. On trial or random lines, the trees will not be 
blazed, unless occasionally, from indispensable neces¬ 
sity, and then it will be done so guardedly as to prevent 
the possibility of confounding the marks of the trial 
line with the true. But bushes and limbs of trees may 
be lopped, and stakes set on the trial or random line, at 
every ten chains, to enable the surveyor on his return 
to follow and correct the trial line and establish there- 



206 


A MANUAL OF LAND SURVEYING. 


from the true line. To prevent confusion, the temporary 
stakes set on the trial or random line will be pulled 
up when the surveyor returns to establish the true 
line. 

Insuperable Objects on Line —Witness 
Points. — 1. Under circumstances where the survey of 
a line is obstructed by an impassable obstacle, such as a 
pond, swamp, or marsh (not meanderable), the line will 
be prolonged across such obstruction by making the 
necessary right-angle offsets; or, if such proceeding is j 
impracticable, a traverse line will be run, or some proper 
trigonometical operation will be employed to locate the 
line on the opposite side of the obstruction; and in case 
the line, either meridional or latitudinal, thus regained, 
is recovered beyond the intervening obstacle, said line 
will be surveyed back to the margin of the obstruction 
and all the particulars, in relation to the field operations, will 
he fully stated in the field notes. 

2. As a guide in alinement and measurement, at each 
point where the line intersects the margin of an obstacle, 
a witness point will be established, except when such point 
is less than 20 chains distant from the true point for a 
legal corner which falls in the obstruction, in which 
case a witness corner will be established at the intersec¬ 
tion. 

3. In a case where all the points of intersection with 
the obstacle to measurement fall more than 20 chains from 
the proper place for a legal corner in the obstruction, 
and a witness corner can be placed on the offset line | 
within 20 chains of the inaccessible corner point, such 
‘•witness corner ” will be established. 

Establishing Corners.— 1 . To procure the faith¬ 
ful execution of this part of a surveyor’s duty, is a mat-1 
ter of the utmost importance. After true coursing and 
most exact measurements, the establishment of corners 
is the consummation of the field work. Therefore, if 
the corners be not perpetuated in a permanent and 
workmanlike manner, the principal object of surveying 
operations will not have been attained. 

2'. The points at which corners will be established are 







ORIGINAL SURVEYS. 


207 


fully stated in the several articles: “Base Lines,” “Prin¬ 
cipal Meridians,” “Standard Parallels, ' etc., following 
the title “Initial Points.” 

3. The best marking tools adapted to the purpose will 
be provided for marking neatly, distinctly, and durably, all 
the letters and figures required to be made at corners, 
arabic figures being used exclusively; and the deputy 
will always have at hand the necessary implements for 
keeping his marking irons in perfect order. 

Descriptions of Corners. — 1 . The form and lan¬ 
guage used in the following articles, in describing, for 
each one of the thirteen classes of corners, eight specific 
constructions and markings, with the stated modifica¬ 
tions in certain cases, will be carefully followed by 
deputy surveyors in tlieir field notes; and their field work 
will strictly comply with the requirements of the 
descriptions. 

2. When pits and mounds of earth are made accesso¬ 
ries to corners, the pits will always have a rectangular 
plan; while the mounds will have a conical form, with 
circular base; and in all cases both pits and mounds will 
have dimensions at least as great as those specified in the 
descriptions. Deputy surveyors will strictly adhere to 
these provisions, and no departure from the stated re¬ 
quirements will be permitted, either in instructions or 
practice in the field. 

3. Referring to the numbered paragraphs, the corners 
described in “3 ” will be preferred to those described in 
either “1” or “2,” when corners are established in 
loose, sandy soil, and good bearing trees are available; 
under similar conditions, the corners described in “5” 
and “8” will be preferred to those described in “4” 
and “7,” respectively. 

4. The selection of the particular construction to be 
adopted in any case will be left, as a matter of course, to 
the judgment and discretion of the deputy, who will as¬ 
sign the greatest weight to the durability of the corner 
materials and permanency of the finished corners. 

5. The following abbreviations and contractions will 
be used in the descriptions of corners, viz.: — 




208 A MANUAL OF LAND SURVEYING. 


A. M. C. for auxiliary meander cor. N. for north. 

bdy. for boundary. % sec. cor. for quarter section corner. 

bdrs. for boundaries. R. for range. 


bet. 

for between. 

C. C. 

for closing corner. 

cor., cors. 

for corner, corners. 

dist. 

for distance. 

E. 

for east. 

ft. 

for foot or feet. 

fracl. 

for fractional. 

ins. 

for inches. 

diam. 

for diameter. 

Iks. 

for links. 

M. C. 

for meander corner. 


Rs. for ranges. 

sec . secs, for section, sections. 

S. M. C. for special meander cor. 

S. C. for standard corner, 

sq. for square. 

S. for south. 

T. or Tp. for township. 

Ts. orTps. for townships. 

W. for west. 

W. C. for witness corner. 

W. P. for witness point. 


For “18 inches long, 7 inches wide, 6 inches thick,’’ 
in describing a corner stone, write ‘'18x7x0 ins.,” be¬ 
ing particular to always preserve the same order of 
length, width, and thickness (or depth), and use a simi¬ 
lar form when describing pits. 


STANDARD TOWNSHIP CORNERS. 

METHOD OF MARKING. 

When more than one half of all the standard town¬ 
ship and section corners on any 6 miles of a base line or 
standard parallel are stone corners, the descriptions in 
paragraphs 1 and 2, if the corners therein described are 
established, will be modified as follows: Strike out “S. 
C., on N.” After “marked,” insert the words: — 

“S. C., 13 N. on N., 

22 E. on E., and 
21 E. on W. faces; ” 

When under the conditions above specified, the corner 
described in paragraph 1 is established, a stake may be 
driven in the east pit and marked instead of the stone, 
and described as exemplified in the last clause of para¬ 
graph 6. 

1. Stone, with Pits and Mound of Earth. 

Set a — stone, —X—X—ins.,— ins. in the ground, for 
standard cor. of (e. g.) Tps. 13 N., Its. 21 and 22 E., 
marked S. C. on N.; with 6 grooves on N., E., and W 
faces; dug pits, 30x24x12 ins., crosswise on each line, E. 
and W., 4 ft., and N. of stone, 8 ft. dist.; and raised a 
mound of earth, 5 ft. base, 2£ ft. high, N. of cor. The 





ORIGINAL SURVEYS. 


209 


direction of .tlie mound, from the corner, will be stated 
wherever a mound is built. 

2. Stone, with Mound of Stme. 

Set a — stone, —X—X— ins., — ins. in the ground, for 
standard cor. of [e. g) Tps. 13 N., Rs. 21 and 22 E., marked 
S. C., on N.; with 6 grooves on N., E., and W. faces; and 
raised a mound of stone, 2 ft. base, 1£ ft. high, N. of 
cor. Pits impracticable. Mound of stone will consist 
of not less than four stones, and will be at least 1£ ft. 
high, with 2 ft. base. 

3. Stone, with Bearing Trees. 

Seta — stone, —X—X— ins., — ins. in the ground, 
for standard cor. of (e. g.) Tps. 13 N., Rs. 21 and 22 E., 
marked S. C., on N.; with 6 grooves on N., E., and W. 
faces; from which 

A —, — ins. diam., bears N. — ° E., — Iks. dist., 
marked 

T. 13 N., R. 22 E., S. 31, B. T. 

A —, — ins. diam., bears N. —° W., — Iks. dist., 
marked 

T. 13 jN\, R. 21 E., S. 36, B. T. 

All bearing trees, except those referring to quarter 
section corners, will be marked with the township, range, 
and section in which they stand. 

4. Post, with Pits and Mound of Earth. 

Set a — post, 3 ft. long, 4 ins. sq., with marked stone 
(charred stake or quart of charcoal), 24 ins. in the 
ground, for standard cor. of ( e. g.) Tps. 13 N., Rs. 22 and 
23 E., marked 

S. 0., T. 13 N. on N. 

R. 23 E., S. 31 on E., and 

R. 22 E., S. 36 on W. faces; with 6 grooves on N., E., 
and W. faces; dug pits, 30x24x12 ins., crosswise on each 
line, E. and W., 4 ft., and N. of post, 8 ft. dist.; and 
raised a mound of earth, 5 ft. base, 2£ ft. high, N. of cor. 

5. Post, with Bearing Trees. 

Set a — post, 3 ft. long, 4 ins. sq., 24 ins. in the ground, 
for standard cor. of (e. g.) Tps. 18 N., Rs. 22 and 23 E., 
marked 


15 






210 


A MANUAL OF LAND SURVEYING. 


S. C., T. 13 K on N., 

R. 23 E. S. 31 on E., and 

R. 22 E., S. 36 on W. faces; with 6 grooves on !N., E., j 
and W. faces, from which 

A —, — ins. dianu, hears N. —° E., — Iks. dist., 
marked 

T. 13 N., R. 23 E., S. 31, B. T. 

A —, — ins. dianu, bears N. —° W., — Iks. dist., 
marked 

T. 13 N., R. 22 E., S. 36, B. T. 

6. Mound of Earth, with Deposit, and Stake in Pit. 

Deposited a marked stone (charred stake or quart of 
charcoal) 12 ins. in the ground, for standard cor. of (e. g.) 
Tps. 13 N., Rs. 22 and 23 E.; dug pits, 30X24X12 ins., 
crosswise on each line, N., E., and W. of cor., 5 ft. dist.; 
and raised a mound of earth, 5 ft. base, 24 ft. high, over 
deposit. 

In E. pit drove a — stake, 2 ft. long, 2 ins. sq., 12 ins. 
in the ground marked 

S. C., T. 13 N. on N., 

R. 23 E., S. 31 on E., and 

R. 22 E., S. 36 on W. faces; with 6 grooves on N., E., 
and W. faces. 

7 Ti •ee Comer, with Pits and Mound of Earth. 

A —, — ins. dianu, for standard cor. of (e. y.) Tps. 13 
!N., Rs. 22 and 23 E., I marked 

S. C., T. 13 1ST. on N., 

R. 23 E., S. 31 on E., and 

R. 22 E., S. 36 on W. sides; with 6 notches on N., E., 
and W. sides; dug pits, 24x18x12 ins., crosswise on each 
line, N., E., and W. of cor., 5 ft. dist.; and raised a 
mound of earth around tree. 

8. Tree Corner, with Bearing Trees. 

A —, — ins. dianu, for standard cor. of (e. g.) Tps. 13 
N., Rs. 22 and 23 E., I marked 

S. C., T. 13 N. on K, 

R. 23 E., S. 31 on E., and 

R. 22 E., S. 36 on W. sides; with 6 notches on N., E., 
and W. sides; from which 





ORIGINAL SURVEYS. 


211 


A —, — ins. diam., bears N. —° E., — Iks. dist., 
marked 

T. 13 N., R. 23 E., S. 31, B. T. 

A —, — ins. diam., bears N. —° W., — Iks. dist., 
marked 

T. 13 N., R. 22 E., S. 36, B. T. 

Witness Corners. — 1. When the true point for 
any corner described in these instructions falls where 
prevailing conditions would insure its destruction by 
natural causes, a witness corner will be established in a 
secure position, on a surveyed line if possible, and within 
twenty chains of the corner point thus witnessed. 

2. Markings on Witness Corners. 

A witness corner will bear the same marks that would 
be placed upon the corner for which it is a witness, and 
in addition, will have the letters “ W. C. ” (for witness 
corner), conspicuously displayed above the regular mark¬ 
ings; such witness corners will be established, in all other 
respects, like a regular corner. 

3. Markings on Bearing Trees of Witness Corners. 

When bearing trees are described as accessories to a 
witness corner, the prescribed markings on each tree 
will be preceded by the letters “W. C.,” distinctly cut 
into the wood. 

The true bearing and distance of witness corners, 
from the true point for the corner, will always be clearly 
stated in the field notes. 

4. Witness Corners to Corner Points Falling in Roads, etc. 

The point for a corner falling on a railroad, street, or 

wagon road, will be perpetuated by a marked stone 
charred stake or quart of charcoal, deposited 24 inches 
in the ground, and witnessed by two witness corners, one of 
which will be established on each limiting line of the 
highway. The deposit will not be practicable in the 
case of railroads ; but the witness corners will be estab¬ 
lished on the lines limiting the right of way. 

In case the point for any regular corner falls at the 
intersection of two or more streets or roads, it will be per¬ 
petuated by a marked stone (charred stake or quart of 




212 


A MANUAL OF LAND SURVEYING. 


charcoal), deposited 24 inches in the ground, and wit¬ 
nessed by two witness corners established on opposite sides 
of the corner point, and at the mutual intersections of 
the lines limiting the roads or streets, as the case may be. 

Witness Points will be perpetuated by corners 
similar to those described for quarter section corners, 
with the marking “ W. P.” (for witness point), in place 
of “4,” or “is.”, as the case may be. 

If bearing trees are available as accessories to witness 
points , each tree will be marked W. P. B. T. (See “In¬ 
superable objects on line — Witness Points.” 

Miscellaneous. — 1. Corners on Bock in Place, or on 
Boulders. 

When a corner falls on rock in place , or on a boulder, a 
cross (X), will be made at the exact corner point, and 
witnessed by the proper number of bearing trees, if they 
are available; in the absence of suitable trees, a mound 
of earth will be raised, if size of the boulder or form of 
the rock in place permits the excavation of pits. As a 
last resort, a mound of stone will be built to attract at¬ 
tention to the point, if loose rock can be obtained in the 
vicinity. 

2. Location of Mounds. 

When mounds of earth or other material are raised 
as accessories to corners, they will be placed as specilied 
in the foregoing Description of Corners, and in every 
case the direction of the mound from the corner will be 
carefully stated. The use of the indefinite description 
“alongside ” will be discontinued. 

In case the character of the land is such that the 
mound cannot be placed as hereinbefore described, the 
deputy will state in his notes, by bearing and distance, 
exactly where the mound is located with reference to 
the corner, and will give his reasons for placing it as 
described. 

3. Mounds of Stone, Covered with Earth. 

In a case where pits are practicable and the deputy 
prefers raising a mound of stone, or a mound of stone 
covered with earth, he will use the form given for 
“ Stone with mound of stone,” when the corner thus de- 



ORIGINAL SURVEYS. 


213 


scribed is established; but when the corner “ Stone, with 
mound of stone covered with earth, ’ ’ is constructed, the de¬ 
scription will be modified as follows : Strike out the 
words “Pits impracticable;” in place of “Mound of 
stone, 2 ft. base, 1£ ft. high,” write “ Mound of stone 
covered with earth, — ft. base, — ft. high,” inserting 
in the blank spaces the dimensions of the mound given 
in paragraph 1, following the designation of each class 
of corners. 

4. Bearing Trees. 

Bearing trees marked as accessories to standard cor¬ 
ners, either township, section, or quarter section, will 
be selected on the north side of base lines or standard 
parallels, and bearing trees referring to the closing cor¬ 
ners on said lines, will be located on the south side; in 
general, the bearing trees referring to any particular 
closing corner, together with one pit and the mound be¬ 
longing to such corner, will be located on the same side of 
the line closed upon, and on the side from which the surveys 
have been closed. 

When the requisite number of trees can be found 
within 300 links of the corner point, two (2) bearing 
trees will be marked and described for every standard 
or closing township or section corner, or corner common 
to two townships or sections, only; four (4) for every 
corner common to four townships or four sections; one 
(1) for a corner referring to one township or one section, 
only; two (2) for every quarter section corner or meander 
corner, and four (4) for each mile or half mile corner, or 
corner monument on a reservation or other boundary, 
not conforming to the system of rectangular surveying. 

In case the prescribed number of trees cannot be 
found within limits, the deputy will state in his field 
notes, after describing those marked, “No other trees 
within limits,” and add “Dug pits—X—X—ins.,’’etc., 
or “ liaised a mound of stone, — ft. base, — ft. high, — 
of cor.,” as prevailing conditions may require. 

Bearing trees, being the most important accessories 
to the corners, will have their exact bearings from the 
true meridian taken with the instrument used in run- 


214 


A MANUAL OP LAND SURVEYING. 


ning the lilies of survey; and the distance from the mid¬ 
dle of each bearing tree to the middle point of the corner will be 
carefully measured, and recorded in the held notes. 

A plain blaze will be made at the usual or most con¬ 
venient height, on each bearing tree, on the side facing 
the corner. The height of all other markings on the 
tree will in no case exceed the limit of two and one half 
feet above the ground. 

5. Stones for Corners. 

Stones 18 ins. long, or less, will be set with two thirds 
of their length in the ground, and those more than 18 
ins. long will have three fourths of their length in the 
ground. 

No stones measuring less than 504 cubic inches, or less 
than 12 ins. in length, will be used for corners. 

6. Objects to be Noted. 

Particular attention is directed to the “Summary of 
objects and data required to be noted,” and the deputy 
will thoroughly comply with the same in his work and 
field notes. 

7. Lines Discontinued at Legal Corners. 

No mountainous lands, or lands not classed as survey- 
able, will be meandered, and all lines approaching such 
lands will be discontinued at the section or quarter-sec¬ 
tion corner nearest the unsurveyed land. 

8. Marks to be cut. 

AH letters and figures on posts, trees, or stones, etc., 

will be cut into the object upon which they are placed. 
Arabic figures and plain letters will be used for all 
markings. 

9. Orientation of Corners. 

Corners referring to one, two, or four townships or 
sections, not identical with standard or closing corners, 
will be set with their faces directed NE. and SW., and 
NW. and SE., while all other corners will be set with 
their sides facing the cardinal points; except corners on 
boundaries of reservations and private land claims, 
which will be set squarely on line. 

10. Size of Posts, Mounds, etc. 






ORIGINAL SURVEYS. 


215 


The sizes of wooden posts, mounds, and pits, noted in 
the foregoing- descriptions, will be regarded as minimum , 
and their dimensions will be increased whenever prac¬ 
ticable. 

11. Corner Materia Is. 

In establishing corners, durable stones will be used 
when obtainable; then, posts; and lastly, mounds, with 
stake in pit. 

Wood of a perishable nature will not be used for posts 
or stakes. 

12. Instructions will be Examined. 

Deputy surveyors will carefully read, study, and fa¬ 
miliarize themselves with all instructions contained in 
this volume, and will instruct their assistants as to 
their duties before commencing work. An extra copy 
of this Manual may be furnished each deputy, for the 
use of his assistants. 

Initial Points. — Initial points from which the lines 
of the public surveys are to be extended will be estab¬ 
lished whenever necessary, under such special instruc¬ 
tions as may be prescribed in each case by the Commis¬ 
sioner of the General Land Office. The locus of such 
initial points will be selected with great care and due 
consideration for their prominence and easy identifica¬ 
tion, and must be established astronomically. 

The lines of the public surveys are classified as fol¬ 
lows:— 

Class 1. Base lines and standard parallels. 

Class 2. Principal and guide meridians. 

Class 3. Township exteriors (or meridional and lati¬ 
tudinal township boundaries). 

Class 4. Subdivision and meander lines. 

The initial point having been established, the line of 
the public surveys will be extended therefrom, as fol¬ 
lows:— 

' Base Line. — 1. From the initial point the base line 
will be extended east and west on a parallel of latitude, 
by the use of transit or solar instruments, as may be 
directed by the surveyor general in his written special 


216 


A MANUAL OF LAND SURVEYING. 


instructions. Tlie transit should be designated for the 
alinement of all important lines. 

2. The direction of base lines will conform to parallels 
of latitude and will be controlled by true meridians; 
consequently the correct determination of true merid¬ 
ians by observations on Polaris at Elongation is a matter of 
prime importance. 

3. When transits are employed, certain reference lines 
having a known position and relation to the required 
parallel of latitude will be prolonged as straight lines, 
by two back and two fore sights at each setting of the 
instrument, the horizontal limb being revolved 180° in 
azimuth between the observations. 

4. Where solar apparatus is used, the deputy will test 
the instrument, whenever practicable, by comparing its 
indications with a meridian determined by Polaris ob¬ 
servations; and in all cases where error is discovered, he 
will make the necessary corrections of his line before 
proceeding with the survey. All operations W'ill be 
fully described in the field notes. 

5. The proper township, section, and quarter section 
corners will be established at lawful intervals, and me¬ 
ander corners at the intersection of the line with all 
meanderable streams, lakes, or bayous. 

6. In order to detect errors and insure accuracy in 
measurement, two sets of cliainmen will be employed; 
one to note distances to intermediate points and to lo¬ 
cate topographical features, the other to act as a check. 
Each will measure 40 chains, and the proper corner will 
be placed midway between the ending points of the two 
measurements. 

The deputy will be present when said corner is thus 
established, and will record in the body of his field notes 
the distances to the same, according to the measure¬ 
ment by each set of cliainmen. 

To obviate collusion between the sets of cliainmen, 
the second set should commence at a point in advance 
of the beginning corner of the first set, the initial dif¬ 
ference in measurement thus obtained being known 
only to the deputy. 







ORIGINAL SURVEYS. 


217 


Principal Meridian. — 1 . This line shall conform to 
a true meridian and will be extended from the initial 
point, either north or south, or in both directions, as 
the conditions may require, by the use of transit or solar 
instruments, as may be directed by the surveyor general 
in his special written instructions. 

2. The methods used for determination of directions, 
and the precautions to be observed to secure accuracy 
in measurement, are fully stated above under the title 
‘‘Base Line,” and will be complied with in every partic¬ 
ular. 

3. In addition to the above general instructions, it is 
required that in all cases where the establishment of a 
new principal meridian seems to be necessary to the 
surveyor general, he shall submit the matter, together 
with his reasons therefor, to the commissioner of the 
General Land Office, and the survey of such principal 
meridian shall not be commenced until written author¬ 
ity, together with such special instructions as he may 
deem necessary, shall have been received from the com¬ 
missioner. 

Standard Parallels.— 1. Standard parallels, which 
are also called correction lines, shall be extended east 
and west from the principal meridian, at intervals of 
every 24 miles north and south of the base line, in the 
manner prescribed for running said line, and all require¬ 
ments under the title “Base Line” will be carefully 
observed. 

2. Where standard parallels have been placed at inter¬ 
vals of 30 or 36 miles, regardless of existing instructions, 
and where gross irregularities require additional stand¬ 
ard lines, from which to initiate new, or upon which 
to close old surveys, an intermediate correction line 
should be established to which a local name may be 
given, e. y., “Cedar Creek Correction Line;” and the 
same will be run, in all respects, like the regular stan¬ 
dard parallels. 

Guide Meridians. — 1. Guide meridians shall be 
extended north from the base line, or standard parallels, 
at intervals of every 24 miles east and west from the 


218 


A MANUAL OF LAND SURVEYING. 


principal meridian, in the manner prescribed for run¬ 
ning the principal meridian, and all the provisions for 
securing accuracy of alinement and measurement found, 
or referred to under the title “Principal Meridian,” 
will apply to the survey of said guide meridians. 

2. When existing conditions require that such guide 
meridians shall be run south from the base or correction 
lines, they will be initiated at properly established clos¬ 
ing corners on such lines. 

3. Where guide meridians have been improperly 
placed at intervals greatly exceeding the authorized 
distance of 24 miles, and standard lines are required to 
limit errors of old, or govern new surveys, a new guide 
meridian may be run from a standard, or properly estab¬ 
lished closing corner, and a local name may be assigned 
to the same, e. g., “Grass Valley Guide Meridian.” 
These additional guide meridians will be surveyed in all 
respects like the regular guide meridians. 

Township Exteriors. — 1. Whenever practicable, 
the township exteriors in a tract of land 24 miles square, 
bounded by standard lines, will be surveyed successively 
through the block, beginning with those of the south¬ 
western township. 

2. The meridional boundaries of townships will have 
precedence in the order of survey and will be run from 
south to north on true meridians, with permanent corners 
at lawful distances; the latitudinal boundaries will be 
run from east to west on random or trial lines, and cor¬ 
rected back on true lines. 

The falling of a random, north or south of the town¬ 
ship corner to be closed upon, will be carefully meas¬ 
ured, and, with the resulting true return course, will be 
duly recorded in the held notes. 

Should it happen, however, that such random inter¬ 
sects the meridian of the objective corner, north or 
south of said corner, or falls short of, or overruns the 
length of the south boundary of the township by more 
than three chains (due allowance being made for conver- 
gency), said random, and, if necessary, all the exterior 


ORIGINAL SURVEYS. 219 

boundaries of the township, will be retraced and re¬ 
measured to discover and correct the error 

When running random lines from east to west, tem¬ 
porary corners will be set at intervals of 40.00 chains, 
and proper permanent corners will be established upon 
the true line, corrected back in accordance with these 
instructions, thereby throwing the excess or deficiency 
against the west boundary of the township, as required 
by law. 

3. Whenever practicable, the exterior boundaries of 
townships belonging to the west range, in a tract or 
block 24 miles square, will first be surveyed in succes¬ 
sion, through the range, from south to north; and in a 
similar manner, the other three ranges will be surveyed 
in regular sequence. 

4. In cases where impassable objects occur and the forego¬ 
ing rules cannot be complied with, township corners will 
be established as follows: — 

In extending the south or north boundaries of a town¬ 
ship to the west, where the southwest or northwest corners 
cannot be established in the regular way by running a 
north and south line, such boundaries will be run west 
on a true line, allowing for convergency on the west half 
mile; and from the township corner established at the 
end of such boundary, the west boundary will be run 
north or south, as the case may be. In extending south or 
north boundaries of a township to the east, where the 
southeast or northeast corner cannot be established in the 
regular way, the same rule will be observed, except that 
such boundaries will be run east on a true line, and the 
east boundary run north or south, as the case may be. 

5. Allowance for the convergency of meridians will be 
made whenever necessary. 

Method of Subdividing. — 1. The exterior bound¬ 
aries of a full township having been properly estab¬ 
lished, the subdivision thereof will be made as follows: — 

At or near the southeast corner of the township, a true 
meridian will be determined’ by Polaris or solar observa¬ 
tions, and the deputy’s instrument will be tested there- 


220 


A MANUAL OF LAND SURVEYING. 


on; then from said corner the first mile of the east and 
south boundaries will be retraced, if subdivisions and 
survey of the exteriors have been provided for in separate 
contracts; but, if the survey of the exterior and subdi- 
visional lines are included in the same contract, the 
retracements referred to will be omitted. All discrep¬ 
ancies resulting from disagreement of bearings or meas¬ 
urements will be carefully stated in the field notes. 

2. After testing his instrument on the true meridian 
thus determined, the deputy will commence at the cor¬ 
ner to sections 35 and 36, on the south boundary, and 
run a line parallel to the range line, establishing at 40.00 
chains, the quarter section corner between sections 35 
and 36, and at 80.00 chains the corner for sections 25, 26, 
35, and 36. 

3. From the last-named corner, a random line wall be 
run eastward, without blazing, parallel to the south bound- 
aip of section 36, to its intersection with the east bound¬ 
ary of the township, placing at 40.00 chains from the 
point of beginning, a post for temporary quarter section 
corner. If the random line intersects said township 
boundary exactly at the corner for sections 25 and 36, it 
will be blazed back and established as the true line, the 
permanent quarter section cprner being established 
thereon, midway between the initial and terminal sec¬ 
tion corners. 

If, however, the random intersects said township 
boundary to the north or south of said corner, the fall¬ 
ing will be carefully measured, and from the data thus 
obtained, the true return course will'be calculated, and 
the true line blazed and established, and the position of 
the quarter section corner determined, as directed 
above. 

The details of the entire operation will be recorded in 
the field notes. 

4. Having thus established the line between sections 
25 and 36; from the corner for sections 25, 26, 35 and 36, 
thewesfand north boundaries of sections 25, 24, 13, and 
12, will be established as directed for those of section 
36; with the exception that the random lines of said 



ORIGINAL SURVEYS. 


221 


north boundaries will be run parallel to the established south, 
boundaries of the sections to which they belong, instead of the 
south boundary of section 36; e. g. the random line be¬ 
tween sections 24 and 25 will be run parallel to the es¬ 
tablished south boundary of section 25, etc. 

5. Then, from the last established section corner, i. e. 
the corner for sections 1, 2, 11, and 12, the line between 
sections J and 2, will be projected northward, on a 
random line, parallel to the east boundry of the township, 
setting a post for temporary quarter section corner at 
40.00chains, to its intersection with the north boundary 
of the township. If the random intersects said north 
boundary exactly at corner for sections 1 and 2, it will 
be blazed back and established as the true line, the 
temporary quarter section corner being established per¬ 
manently in its original position, and the fractional 
measurement thrown into that portion of the line be¬ 
tween said corner and the north boundary of the town¬ 
ship. 

If, however, said random intersects the north bound¬ 
ary of the township, to the east or west of the corner 
for sections 1 and 2, the consequent falling will be care¬ 
fully measured, and from the data thus obtained, the 
true return course will be calculated, and the true line 
established; the permanent quarter section corner being 
placed upon the same at 40.00 chains from the initial 
corner of the random line, thereby throwing the frac¬ 
tional measurement in that portion lying between the 
quarter section corner and the north boundary of the 
township. 

When the north boundary of a township is a baseline 
or standard parallel, the line between sections 1 and 2 
will be run parallel to the range line as a true line, the quar¬ 
ter section corner will be placed at 40.00 chains, and a 
closing corner will be established at the point of inter¬ 
section with such base or standard line ; and in such 
case, the distance from said closing corner, to the near¬ 
est standard corner on such base or standard line, 
will be carefully measured and noted as a connection 
line. 


222 


A MANUAL OF LAND SURVEYING. 


6. Each successive range of sections progressing to 
the west, until the fifth range is attained, will be sur¬ 
veyed in a similar manner; then, from the section cor¬ 
ners established on the west boundary of said range of 
sections, random lines will be projected to their inter¬ 
section with the west boundary of the township, and 
the true return lines established as prescribed for the 
survey of the first or most eastern range of sections, 
with the exception that on the true lines thus estab¬ 
lished, the quarter-section corners will be established at 
40.00chains from the initial corners of the randoms, the 
fractional measurements being thereby thrown into 
those portions of the lines situated between said quar¬ 
ter-section corners and the west boundary of the town¬ 
ship. 

7. The following general requirements are reiterated 
for emphasis: — 

The random of a latitudinal section line will always 
be run parallel to the south boundary of the section to 
which it belongs, and with the true bearing of said 
boundary; and when a section has no linear south 
boundary, the random will be run parallel*to the south 
boundary of the range of sections in which it is situ¬ 
ated, and fractional true lines will be run in a similar 
manner. 

8. The deputy is not required to complete the survey 
of the first range of sections from south to north before 
commencing the survey of the second or any subsequent 
range of sections, but the corner on which any random 
line closes shall have been previously established by 
running the line which determines its position, except 
as follows: Where it is impracticable to establish such 
section corner in the regular manner, it will be estab¬ 
lished by running the latitudinal section line as a true 
line, with a true bearing, determined as above directed 
for random lines, setting the quarter-section corner at 
40.00 chains and the section corner at 80.00 chains. 

9. Quarter-section corners, both upon meridional and 
latitudinal section lines, will be established at points 
equidistant from the corresponding Section corners, except 



ORIGINAL SURVEYS. 


223 


upon the lines closing on the north and west boundaries 
of the township, and in those situations the quarter- 
section corners will always be established at precisely 
forty chains to the north or west (as the case may be) of 
the respective section corners from which those lines 
respectively start, by which procedure the excess or de¬ 
ficiency in the measurements will be thrown, according 
to law, on the extreme tier or range of quarter sections, 
as the case may be. 

10. Where by reason of impassable objects only a por¬ 
tion of the south boundary of a township can be estab¬ 
lished, an auxiliary base line (or lines, as the case may 
require) will be run through the portion which has no 
linear south boundary, first random, then corrected, 
connecting properly-established corresponding section 
corners (either interior or exterior) and as far south as 
possible, and from such line or lines, the section lines 
will be extended northwardly in the usual manner, and 
any fraction south of said line will be surveyed in the 
opposite direction from the section corners on the aux¬ 
iliary base thus established. 

11. Whereby reason of impassable objects no portion 
of the south boundary of a township can be regularly estab¬ 
lished, the subdivision thereof will proceed from north 
to south and from east to west, thereby throwing all frac¬ 
tional measurements and areas against the west bound¬ 
ary, and the meanderable stream or other boundary 
limiting the township on the south. 

If the east boundary is without regular section corners 
and the north boundary has been run eastwardly as a 
true line, with section corners at regular intervals of 
80.00 chains, the subdivision of the township will be 
made from west to east, and fractional measurements and 
areas will be thrown against the irregular east bound¬ 
ary. 

12. When the proper point for the establishment of a 
township or section corner is inaccessible, and a witness 
corner can be erected upon each of the two lines which 
approach the same, at distances not exceeding twenty 
chains therefrom, said witness corners will be properly 


224 


A MANUAL OF LAND SURVEYING. 


established, and the half miles upon which they stand 
will be recognized as surveyed lines. 

The witness corner will be marked as conspicuously 
as a section corner, and bearing trees will be used wher¬ 
ever possible. 

The deputy will be required to furnish good evidence 
that the section corner is actually inaccessible. 

Meandering.— 1. Proceeding dmvn stream, the bank 
on the left hand is termed the left bank and that on the 
right hand the right bank. These terms will be univers¬ 
ally used to distinguish the two banks of a river or 
stream. 

2. Navigable rivers, as well as all rivers not embraced 
in the class denominated “navigable,” the right-angle 
width of which is three chains and upward, will be 
meandered on both banks, at the ordinary mean high 
water mark, by taking the general courses and distances 
of their sinuosities, and the same will be entered in the 
field bo'ok. Rivers not classed as navigable will not be 
meandered above the point wdiere the average right- 
angle width is less than three chains. Shallow streams, 
without any well-defined channel or permanent banks, 
will not be meandered, except tide-water streams, whether 
more or less than three chains wide, which should be 
meandered at ordinary high-water mark, as far as tide¬ 
water extends. 

At every point where either standard, township, or 
section lines intersect the bank of a navigable stream, 
or any meanderable line, corners will be established at 
the time of running these lines. Such corners are 
called meander corners, and the deputy will commence 
at one of these corners, follow the bank or boundary 
line, and measure the length of each course from the 
beginning corner to the next “meander corner.” Com¬ 
pass courses, by the needle or solar, will be used in me¬ 
anders. Transit angles are not allowed. 

The crossing distance between meander corners on 
same line and the true bearing and distance between 
corresponding meander corners will be ascertained by 
triangulation, or direct measurement, in order that the 




ORIGINAL SURVEYS. 


225 


river may be protracted with entire accuracy. The par¬ 
ticulars will be given in the held notes. 

In meandering water courses or lakes, where a dis¬ 
tance is more than ten chains between successive sta¬ 
tions, whole chains only should be taken ; but if the 
distance is less than ten chains, and it is found conven¬ 
ient to employ chains and links, the number of links 
should be a multiple often , thereby saving time and labor 
in testing the closings, both in the field and office. 

3. The meanders of all lakes, navigable bayous, and 
deep ponds, of the area of twenty-five acres and up¬ 
wards, will be commenced at a meander corner and con¬ 
tinued, as above directed for navigable streams; from 
said corner, the courses and distances of the entire mar¬ 
gin of the same, and the intersections with all meander 
corners established thereon will be noted. 

All streams falling into the river, lake, or bayou will 
be noted, and the width at their mouths stated; also, 
the position, size, and depth of springs, whether the 
water be pure or mineral; also the heads and mouths of 
all bayous; all islands, rapids, and bars will be noted, 
with intersections, to their upper and lower ends, to es¬ 
tablish their exact situation. The elevation of the 
banks of lakes, bayous, and streams, the height of falls 
and cascades, and the length and fall of rapids will be 
recorded in the field notes. 

To meander a lake or deep pond lying entirely within 
the boundaries of a section, two lines will be run from 
the two nearest corners on different sides of such lake 
or pond, the courses and length of which will be re¬ 
corded, and if coincident with unsurveyed lines of legal 
subdivisions, that fact will also be stated in the field 
notes, and at each of the points where said lines inter¬ 
sect the margin of the pond or lake, a special meander 
corner will be established as above directed. 

The relative position of these points being thus defi¬ 
nitely fixed in the section, the meandering will com¬ 
mence at one of them and be continued to the other, 
noting the intersection, and thence to the beginning. 
The proceedings are to be fully entered in the field notes. 

4. Meander lines will not be established at the segre¬ 
gation line between dry and swamp or overflowed land, 
but at the ordinary hirfh-water mark of the actual margin 
of the rivers or lakes on which such swamp or overflowed 
lands border. 

5. The precise relative position of an island, in a town- 


id 


226 


A MANUAL OF LAND SURVEYING. 


ship made fractional by a river or lake in which the isl¬ 
and is situated, will be determined by triangulation 
from a special and carefully measured base line, initi¬ 
ated from the surveyed lines, on or near the lake or 
river bank on the main land, so as to connect by course ! 
and distance on a direct line, the meander corner on the 
mainland with the corresponding point on the island, 
where the proper meander corner will be established. 

6. In making the connection of an island lying en¬ 
tirely within a section, with the mainland, a special - 
base will be measured from the most convenient mean¬ 
der corner, and from such base, the location of an 
auxiliary meander corner will be determined by triangu¬ 
lation, at which the meanders of the island will be 
initiated. 

7. In the survey of lands bordering on tide water, “me¬ 
ander corners ” will be established at the points where j 
surveyed lines intersect high-water mark , and the mean¬ 
ders will follow the high-water line. 

8. The field notes of meanders will show the dates on 
which the work was performed, as illustrated in the 
specimen notes. The field notes of meanders will state 
and describe the corner from which the meanders 
commenced, and upon which they closed, and will 
exhibit the meanders of each fractional section sepa¬ 
rately; following, and composing a part of such notes, 
will be given a description of the land, timber, depth of 
inundation to which the bottom is subject, and the 
banks, current, and bottom of the stream or body of 
water meandered. The utmost care will be taken to 
pass no object of topography, or change therein, without 
giving a particular description thereof in its proper 
place in the notes of the meanders. 

Summary of Objects and Data Required to 
be Noted. — 1 . Tfie precise length of every line run, 
noting all necessary offsets therefrom, with the reason 
for making them, and method employed. 

2. The kind and diameter of all bearing trees, with 
the course and distance of the same from their respect¬ 
ive corners; and the precise relative position of witness 
corners to the true corners. 

3. The kind of materials of which corners are con¬ 
structed. 

4. Trees on line. The name, diameter, and distance on 
line to all trees which it intersects. 

5. Intersections by line of land objects. The distance 
at which the line intersects the boundary lines of every 
reservation, mining claim, settler’s claim, improvement, 
or rancho; prairie, bottom land, swamp, marsh, grove, 


ORIGINAL SURVEYS. 


227 


and windfall, with the course of the same at all points 
of intersection; also, the distances at which the line 
begins to ascend, arrives at the top, begins to descend, 
and reaches the foot of all remarkable hills and ridges, 
with their courses, and estimated height in feet , above the 
level land of the surrounding country, or above the bot¬ 
tom lands, ravines, or waters near which they are situ¬ 
ated. Also, distance to and across large ravines, their 
depth and course. 

6. Intersections by line of water objects. • All rivers, 
creeks, and smaller streams of water which the line 
crosses; the distances measured on the true line to the 
bank first arrived at , the course down stream at points of 
intersection, and their widths on line. In cases of navigable 
streams, their width will be ascertained between the 
meander corners , as set forth under the proper head. 

7. The land’s surface —whether level, rolling, broken, 
hilly, or mountainous. 

8. The soil —whether first, second, third, or fourth 
rate. 

9. Timber — the several kinds of timber and under¬ 
growth, in the order in which they predominate. 

10. Bottom lands — to be described as wet or dry, and if 
subject to inundation, state to what depth. 

11. Springs of water — whether fresh, sal i ne, or mineral, 
with the course of the stream flowing from them. 

12. Lakes and ponds— describing their banks and giv¬ 
ing their height, and also depth of water, and whether 
it be pure or stagnant. 

13. Improvements.— Towns and villages; houses or cab¬ 
ins, fields, or other improvements with owner’s names; 
mill sites, forges, and factories, mineral monuments, 
and all corners not belonging to the system of rectangu¬ 
lar surveying; will be located by bearing and distance, 
or by intersecting bearings from given points. 

14. Coal banks or beds; peat or turf grounds; minerals 
and ores; with particular description of the same as to 
quality and extent, and all diggings therefor; also salt 
springs and licks. All reliable information that can be 
obtained respecting these objects, whether they be on 
the line or not, will appear in the general description to 
be given at the end of the notes. 

15. Loads and trails, with their directions, whence and 
whither. 

16. Rapids, cataracts, cascades, or falls of water, with 
the estimated height of their fall in feet. 

17. Precipices, caves, sink holes, ravines, stone quar¬ 
ries, ledges of rocks, with the kind of stone they afford. 

18. Natural curiosities, interesting fossils, petrifactions, 


228 


A MANUAL OF LAND SURVEYING. 


organic remains, etc. ; also all ancient works of art, such 
as mounds, fortifications, embankments, ditches, or ob¬ 
jects of like nature. 

19. The magnetic declination will be incidentally noted at 
all points of the lines being surveyed, where any material \ 
change in the same indicates the probable presence of 
iron ores; and the position of such points will be per- j 
fectly identified in the field notes. 

Prescribed Limits for Closings and Lengths 
of Lines. — 1. If in running a random township exter¬ 
ior, such random falls short of or exceeds its proper 
lengths by more than three chains, or falls more than 
three chains north or south of its objective corner, it will 
be re-run, and if found correct, so much of the remain¬ 
ing boundaries of the township will be retraced or re¬ 
surveyed, as may be found necessary to locate the error. 

2. Every meridional section line, except those termi¬ 
nating in the north boundary of the township, shall be 
eighty chains in length. 

3. Tire random meridional section lines through the 
north tier of sections shall fall within fifty links east or 
west of the section corners established on the north 
boundary of the township, except when closing on a base 
line or standard parallel. 

4. Every meridional section line through the north tier 
of sections, shall be within fifty links of the actual dis¬ 
tance established on the east boundary line of the town¬ 
ship for the width of said tier of sections. 

5. All random latitudinal section lines shall fall within 
fifty links north or south of their objective section cor¬ 
ners. 

The latitudinal section lines, except those terminating 
in the west boundary of the township, shall be within 
fifty links of the actual distance established on the south 
boundary line of the township for the width of the 
range of sections to which they belong. 

6. The north boundary and the south boundary of any 
one section, except in the extreme western range of sec¬ 
tions, shall be within fifty links of equal length. 

7. The meanders within each fractional section, or 
between any two successive meander corners, or of an 
island in the interior of a section, should close within a 
limit to be determined by allowing five eighths of a link 
for each chain of said meander line. Where the meander 
corners marking the ends of a meander line in a frac¬ 
tional section are located on standard, township, or sec¬ 
tion lines, the above limit, increased by one fourth of the, 
regular perimeter of the fractional section, expressed in miles , 
multiplied by 71 links, will be allowed. 



ORIGINAL SURVEYS. 


229 


The extreme limit, however, will in no case be per¬ 
mitted to exceed one hundred and fifty links. 

Field Notes. — 1. The proper blank books for origi¬ 
nal field notes will be furnished by the surveyor general, 
and in such books the deputy surveyor will make a 
faithful, distinct, and minute record of everything done 
i and observed by himself and his assistants, pursuant to 
instructions, in relation to running, measuring, and 
marking lines, establishing corners, etc., and present, 
as far as possible, full and complete topographical 
sketches of all standard and exterior lines, drawn to the 
usual scale for township exteriors. These “original 
Held notes ” are not necessarily the entries made in the 
field, in the deputy’s pocket note books called tablets: 
but they are to be fully and correctly written out in 
ink. from such tablets, for the permanent record of the 
work. Tablets should be so fully written as to verify 
the “original Held notes ” whenever the surveyor-gen¬ 
eral requires them for inspection. 

2. A full description of all corners belonging to old 
surveys, from which the lines of. new surveys start , or 
upon which they dose , will in all cases be furnished the 
deputy from the surveyor general’s office, when author¬ 
ity is given for commencing work; then, if the old cor¬ 
ners are found to agree with.said descriptions, the 
deputy will describe any one of them in this form, 
“which is a stone Hrmly set. marked, and witnessed, as 
described by the surveyor general; ” but should a corner 
not answer the description supplied, the deputy will 
give a full description of such corner and its accessories, 
following the proper approved form given in these 

1 instructions. 

A full description of each corner established under 
any one contract will be given once only ; subsequent 
reference to such corner will be made in the form, 
“heretofore described,” or (c. y.) “the corner for sec¬ 
tions 2. 3, 10, and 11,” as the case may require. 

In all cases where a corner is re-established, the original 
field notes will describe fully the manner in which it is 
done. 

3. Th e original field notes of the survey of base, stan¬ 
dard, and meridian lines will describe all corners estab¬ 
lished thereon, how established, the crossings of streams, 
ravines, hills, and mountains; character of soil, timber, 
minerals, etc.; and after the description of each town¬ 
ship corner established in running such lines, the deputy 
will note particularly in the “general description ” the 
character of townships on each side of the lines run. 

4. The original field notes of the survey of exterior 





230 


A MANUAL OF LAND SURVEYING. 


boundaries of townships will describe the corners and 
topography, as above required, and the “general de¬ 
scription ” at the end of such notes will describe the 
townships as fully as possible, and also state whether or 
not they should be subdivided. 

5. The original field notes of the subdi visional survey of 
townships will describe the corners and topography as 
above required, and the “general description ” at the 
end of such notes will state minutely the character of 
the land, soil, timber, etc., found in such townships. 

The topography will be given on the true line in all 
cases, and will be taken correctly, not estimated or 
approximated. 

6. With the original field notes of the survey of base 
lines and standard parallels, and principal and guide 
meridians forming a tract 24 miles square, including 
those of the township exteriors therein, the deputy 
will submit a diagram of the lines surveyed, drawn to a 
scale of half an inch to one mile, upon which will be 
written the true bearings and lengths of all surveyed lines, 
except the lengths of those which are actually 40.00 or 
80.00 chains. These diagrams will exhibit all water 
courses, with the direction of each indicated by an ar¬ 
row head pointing doicn stream; also, the intersection of 
the lines with all prairies, marshes* swamps, ravines, 
lakes, ponds, mountains, hills, and all other natural or 
artificial topographical features mentioned in the origi¬ 
nal field notes , to the fullest extent possible. 

7. With the special instructions for making subdivis- 
ional surveys of townships into sections, the deputy will 
be furnished by the surveyor general with blank township 
diagrams drawn to a scale of one inch to forty chains, upon 
which the true bearings and lengths of the township 
and section lines, from which the surveys are to be pro¬ 
jected, or upon which they are to close, will be carefully 
marked; and on such diagrams the deputy who subdi¬ 
vides will make appropriate sketches of the various ob¬ 
jects of topography as they occur on his lines, so as to 
exhibit not only the points of intersection therewith, 
but also the directions and relative positions of such 
objects between the lines, or within each section, as far 
as practicable, so that every topographical feature may 
be properly completed and connected in the showing. 

8. Triangulations, offsets, or traverses, made to deter¬ 
mine distances that cannot be directly measured, such 
as those over (e. g.) deep streams, lakes, impassable 
swamps, canons, etc., will be made on the random lines, 
when random lines are run. All particulars will be fully 
stated in the field notes. 


SUBDIVISION OF SECTIONS. 


231 


CHAPTER IX. 

SUBDIVISION OF SECTIONS. 

1. Subdivisions of sections are original surveys 
to be made in the following manner: 

1. Section and quarter-section corners set by the gov¬ 
ernment surveyors, and the boundaries actually run by 
them, as well as the length of all lines as returned in 
their field notes, are to be taken as correct. (See Sec. 2396 
R. S., First and Second. P. 182, Sec. 100.) 

2. The corners of half and quarter sections which were 
not marked on the government surveys, must be placed 
as nearly as possible equidistant from those two corners 
which stand on the same line. (Sec. 2396, First. P. 182, 
Sec. 100.) 

This applies to the quarter-posts on the north and west 
lines of the township which were surveyed previous to 
1846; also to those townships which, under the act of 1796, 
were surveyed into blocks of two miles square (P. 180, 
Sec. 99, Third), and to those surveyed under the act of 
1800,* where no quarter-section corners were planted on 
the lines running from south to north. 

*No. 21 .—An Act to amend the act entitled “An act providing for 

the sale of the lands of the United .States, in the territory northwest 

of the Ohio, and above the mouth of the Kentucky River.” 

Sec. 3. And be it further enacted , That the surveyor-general shall 
cause the townships west of the Muskingum, which by the above- 
mentioned act are directed to be sold in quarter townships, to be sub¬ 
divided into half sections of three hundred and twenty acres each, as 
nearly as may be, by running parallel lines through the same from east 
to west, and from south to north, at the distance of one mile from each 






232 


A MANUAL OP LAND SURVEYING. 


3. The boundary lines of sections, (see Page 180, Sec. 99, 
Third), and of half and quarter sections, which were not 
actually run and marked, are to be ascertained by run¬ 
ning straight lines from the established corners to the 
opposite corresponding corners. Where no such opposite 
corners have been or can be fixed, the line should be run 
from the established corner due north and south or east 
and west, as the case may be, to the water-course or 
other external boundary. (P. 182, Sec. 100, Second.) These 
due lines are to be found by trial of the boundary lines 
of the section, as actually run by the government sur¬ 
veyor, and the subdivision line, run on a course interme¬ 
diate between the courses of the section lines which lie 
parallel with it. 

The following figure illustrates the manner of sub¬ 
dividing sections. It shows sections 5, 6, 7, and 8, repre- 


other, and marking corners, at the distance of each half mile on the 
lines running from east to west, and at the distance of each mile on 
those running from south to north, and making the marks, notes, and 
descriptions prescribed to surveyors by the above-mentioned act: And 
the interior lines of townships intersected by the Muskingum, and of 
all the townships lying east of that river, which have not been hereto¬ 
fore actually subdivided into sections, shall also be run and marked in 
the manner prescribed by the said act for running and marking the 
interior lines of townships directed to be sold in sections of six hun¬ 
dred and forty acres each. And in all cases where the exterior lines 
of the townships, thus to be subdivided into sections or half-sections, 
shall exceed or shall not extend six miles, the excess or deficiency 
shall be specially noted, and added to or deducted from the western 
and northern ranges of sections or half-sections in such township, ac¬ 
cording as the error may be in running the lines from east to west, or 
from south to north; the sections and half-sections bounded on the 
northern and western lines of such townships shall be sold as contain¬ 
ing only the quantity expressed in the returns and plats, respectively, 
and all others as containing the complete legal quantity. And the 
President of the United States shall fix the compensation of the dep¬ 
uty surveyors, chain-carriers, and axemen: Provided , The whole ex¬ 
pense of surveying and marking the lines shall not exceed three 
dollars for every mile that shall be actually run, surveyed, and 
marked. 



SUBDIVISION OF SECTIONS. 


233 


senting the four different cases which occur in a township 
surveyed previous to 1846. In the later surveys, the de- 



Case 1 . —Section 8. All the quarter posts are at equi¬ 
distant points from the section corners which are on the 
same line. 

Case 2.—Section 5. Quarter posts on the north and 
the south are at equidistant points. Those on the east 
and the west are 40 chains from the south line of the sec¬ 
tion. The fraction is on the north half of the section. 

Case 3.—Section 7. Quarter posts on the north and 
the south are placed at 40 chains from the east line of the 
section. Those on the east and the west are at equidistant 
points. The west half of the section is fractional. 

Case 4 —Section 6. The quarter posts on the north 
and the south are placed at 40 chains from the east line 
of the section. Those on the east and the west are 40 
chains from the south line of the section. Fractional 
both on the north and west. 

Note.— In 1856, Thomas A. Hendricks, then Commissioner of the 
General Land Office, gave the following rule for locating the center of 
a section: “Bun a true line from the quarter-section corner on the 
east boundary, to that in the west boundary, and at the equidistance 
between them establish the corner for the center of the section.” 






















234 


A MANUAL OF LAND SURVEYING. 


This was in harmony with an opinion previously given by the Sur¬ 
veyor General of Missouri and Illinois, and was very generally fol¬ 
lowed by the surveyors in those States. This rule has not been sus¬ 
tained by the courts, nor by any other ruling of the Land Office, so far 
as we can learn. It was expressly overruled by the Secretary of the 
Interior in 1868. 

Quarter-sections are to be subdivided into half-quar- 
ters by lines running north and south. 

The corners which were not marked are to be placed as 
nearly as possible equidistant between the two corners 
of the quarter-section which stand on the same line. 
Then run straight lines from the established corners to 
the opposite corresponding corners, (Page 183, Sec. 101.) 

Half-quarter sections are to be subdivided into quar¬ 
ter-quarters in a similar manner, by east and-west lines. 
(P. 183, Sec. 101.) 

It may be well to remark here, that the instructions from the Gen 
eral Land Office have not been uniform in regard to the proper manner 
of subdividing quarter-sections, and, as might be expected, the prac¬ 
tice is not uniform among good surveyors Commissioners Wilson 
and Edmunds held t hat half-quarter and quarter-quarter lines should 
be “ straight lines running through the section ” to points on the sec¬ 
tion line. (See Hawes’s Manual, p. 142, and Dunn’s Land Laws, p, 19.) 

The foregoing rules are those of the statute, and y^re endorsed by 
Commissioners Drummond, Williamson, and McFarland. 

Commissioner Drummond’s instructions are as follows: 

“ In the subdivision of quarter-sections, the quarter-quarter posts 
are to be placed at points equidistant, and on straight lines between 
the section and quarter-section corners, and between the quarter-cot •- 
tiers and the common center of the section ” etc. The difference in the 
two methods occurs when, as very often happens, the quarter-posts 
are not in line between the section corners. 

2. Fractional sections are to be subdivided ac¬ 
cording to the Fifth paragraph of Sec. 2395 of the Revised 
Statutes, under such rules and regulations as may be pre¬ 
scribed by the Secretary of the Interior. (Sec. 99, Ex. 
Land Laws, and U. S. Instructions, 1881, p. 39.) 

Under these regulations,* the fractional quarter-sections 
lying next to the north line of the township are divided 

* Note.— “ Circular to Survey ors-Oenerat, Nov. 9 , 1S21. — Sir: By the 
first section of the act of April 24,1820, all the public lands of the Uni¬ 
ted States shall be offered at public sale in half-quarter sections; and 




SUBDIVISION OF SECTIONS. 


235 


into halt-quarters by lines running east and west, parallel 
with and twenty chains distant from the quarter-section 
line. (See Keasling v. Truitt, 30 Ind. 506.) 

The quarter-sections lying next to the west line of the 
township are divided into half-quarters by lines running 
north and south, parallel with and twenty chains distant 
from the quarter-section line. 

3. Section 6 adjoins both the north and the west 
lines of the township, and is subject to both rules. The 
north half is divided into half quarters by an east and 
west line, and the south half by north and south lines. 

The quarter-post on the north side of section six should 
be placed on the township line at a point 40 chains of 
original measure west from the northeast corner of the 
section. 

The quarter-post on the west line of section six should 
be placed at a point on the range line 40 chains of orig¬ 
inal measure north from the southwest corner of the 
section. By original measure is meant such measure as 
was actually laid down on the ground by the deputy sur¬ 
veyors who made the original survey. 


fractional sections containing one hundred and sixty acres and up¬ 
ward shall, as nearly as practicable, he divided into half-quarter sec¬ 
tions, under such rules and regulations as may be prescribed by the 
Secretary of the Treasury; hut fractional sections containing less than 
one hundred and sixty acres shall not he divided, etc. By the act of 
May 10,1800, section 3, the excess or deficiency of regular sections or 
quarter-sections in any township is to be thrown on the north and 
west sides of the township, making fractional sections more or less 
than one hundred and sixty acres. In subdividing such fractional 
sections to form a half-quarter section, viz., 80 acres, the Secretary of 
the Treasury directs that the subdividing line for such fractions as lie 
on the north side of a township shall be an east and west line, forming 
the half-quarter section on the south side of the fraction; and for such 
fractions as lie on the west side, the subdividing line shall be a merid¬ 
ian, forming the half-quarter section on the east side of the fraction. 
This mode of subdivision will preserve the compactness of the tracts 
with the general divisions, and will not interfere with the rule adopted 
relative to fractions formed by a stream, a river, etc.” 



233 


A MANUAL OF LAND SURVEYING. 


Iii further subdividing the northwest quarter of Section 
6 into quarter-quarters, it is done by a line parallel with 
and 20 chains west of the north and south quarter section 
line. 

The foregoing is the general plan adopted for the sub¬ 
division of sections of the United States Survey. There 
have, however, been many exceptions in the earlier official 
plats, in accordance with which the land was sold. To 
meet all such cases the rule has been adopted to subdivide 
in such a way as to suit the calculation of the areas on 
the official plat. This is sometimes difficult, the areas in 
some cases seeming to have been put down without any 
calculation. 

Sections made fractional by waters, reser¬ 
vations, e t c., 
should be sub¬ 
divided in such 
a manner as to 
produce the 
same result as 
would have 
been produced 
had the section 
been full. This 
may sometimes 
be done by ex¬ 
tending and by 
measuring the 
lines on the ice, 
or over the res¬ 
ervation. 


Commissioner Drummond says (see Copp’s Land Laws, 
p. 7f>l): “In the subdivision of fractional sections, where no 
opposite corners have been or can be fixed, the subdivision 
lines should be ascertained by running lines from the es¬ 
tablished corners due north, south, east or west, as the case 


Fig. to. 



Figure illustrating the Subdivision of a Section 
fractional on waters. 










SUBDIVISION OF SECTIONS. 


237 


may be, to the water-course, Indian boundary line, or 
other external boundary line of such fractional section. 
The law presupposes the section lines surveyed and 
marked in the field by the United States deputy survey¬ 
ors to be due north and south or east and west lines. 
But in actual experience, this is not always the case. 
Hence, in order to carry out the spirit of the law, it will 
be necessary in the running of subdivisional lines through 
fractional sections to adopt mean courses where the lines 
are not due lines, or to run the subdivisional line paral¬ 
lel with the section line when there is no opposite section 
line.” 

4. Irregular Subdivisions of Fractional Sec¬ 
tions.—In making irregular subdivisions of fractions 
bounded on streams or lakes, there seems to have been 
no rule laid down by the authorities. 

It has been decided by the Supreme Court of the United 
States that “the meander lines run in surveying frac¬ 
tional portions of the public lands bordering upon navi¬ 
gable rivers are run not as boundaries of the tract but 
for the purpose of defining the sinuosities of the stream 
and as the means of ascertaining the quantity of land in 
the fraction, and which is to be paid for by the pur¬ 
chaser.” 

R. R. Co. v, Sclmrmier, 7tli Wallace (U.S.) 272. 

It is fair to infer that the same lines are to be used in 
ascertaining the quantity of land in any portion of the 
fraction. Thus, as often happens, if a deed calls for so 
many acres off the end of the fraction, the surveyor in 
making his computations to determine at what point to 
locate the dividing line, should in the absence of any¬ 
thing showing to the contrary, use the meander line for 
the purpose of estimating the area of the tract, and lay 
down the dividing line accordingly. Otherwise there 
could be no common basis of calculation and as many 
different results would be arrived at as there were differ¬ 
ent surveyors to run the line, or different times of survey. 


238 


A MANUAL OF LAND SURVEYING. 


This is especially true of fractions bordering on lakes 
whose shore lines are subject to great change from natu¬ 
ral causes or artificial drainage. 

The common law rule for calculating the quantity of 
land bordering on a non-navigable stream is that no ref¬ 
erence is had to what lies between low water mark and 
the centre of the stream. On navigable waters, high 
water mark is the line. 

Lamb v. Rickett, 11 Ohio 311. 

5 t Exceptional Oases.—In the United States sur¬ 
veys made previous to 1815, there was much irregularity 
in the practice of the surveyors in carrying on the sur¬ 
veys. The fractional sections were frequently thrown 
upon the south or east tiers of sections in the township; 
the surveys being carried on from the north to the south 
and from the west to the east. Where the township was 
made fractional by large rivers or lakes, they were fre¬ 
quently so laid off as to throw all the fractions into the 
sections bordering on the water. 

There was even greater irregularity in the manner of 
subdividing the fractional sections into the lesser tracts. 
Many of them had no quarter section corners. In some, 
the government plats show no subdivision; some are sub¬ 
divided in one way and some in another. 

In making resurveys and subdivisions of these and all 
other exceptional cases, the surveyor must always make 
his resurvey conform to the plan as shown by the field- 
notes and plats of the original survey. 

6. Other Original Surveys.—In a considerable 
portion of the United States, the general government 
never had any ownership of the land. 

The surveys were there made by the proprietors upon 
such system or plan as suited themselves. 

The further subdivision of these tracts is original sur¬ 
veying. It is sufficient to say of this work that it should 
be done with great care, and that the marks upon the 
ground which indicate the boundary lines should bo of 


SUBDIVISION OF SECTIONS. 239 

the plainest and most permanent character which the 
circumstances of the case permit, 

These marks are intended to fix for all time the boun¬ 
daries of the tract laid off and they cannot be too plain or 
permanent. Want of due care and precaution in making 
permanent land marks upon the ground, at the time of 
the original survey, is the fruitful cause from which 
arises most of the litigation about boundary lines. 

7 . Highway surveys, like other surveys, lose much 
of their value if their corners and lines are not so thor¬ 
oughly marked as to be readily found at any future time. 
The centre line of a highway is very commonly used as a 
boundary line. Good permanent landmarks, well guarded 
by bearings and distances to the most permanent objects 
in the vicinity should be planted at the starting and 
closing points of the surve} r , at each angle in its course, 
and at every crossing of a section line. The distance of 
the crossing points should be given from the nearest gov¬ 
ernment corners each way on the section line. 

8. Surveys for town plats are made upon any 
system to suit the circumstances of the case, or the views 
of the owners of the land platted. 

In making these surveys, it is important that the work 
be in every respect carefully done; that full and complete 
notes be taken, so that the plat when finished shall show 
every material fact which may be of use to the public or 
to the future surveyor. 

The relation which the lines of the plat bear to the 
lines of the original boundaries, whether of the govern¬ 
ment survey or otherwise, should be shown on the plat, 
and, what is most important of all, the location of the 
lines upon the ground should be marked by a sufficient 
number of permanent monuments so that there may never 
arise any difficulty in determining the exact position 
those lines occupy. 

Such monuments should be placed at the corners and 
angles of the tract platted, and if included in the United 


240 


A MANUAL OF LAND SURVEYING. 


States survey, they should be placed at the corners of the 
legal subdivisions of a section which are included in the 
plat. Monuments should also be placed so as to define 
the lines and termini of all streets. 

For this purpose, they may be placed either along the 
centre lines and angles of the streets or along their mar¬ 
gins at the corners and angles of blocks. Each method 
has its advantages and disadvantages. The surveyor 
should consider the special circumstances of each case, 
and so locate the monuments that, while effecting the 
purpose for which they were intended, they shall be 
the most likely to remain in position and the easiest to 
refer to. 

9. In Michigan town plats are required by law (Session Laws of 
1885) to be made and recorded in the following manner: 

The plats must be made on sheets of good muslin backed paper, 18 
inches by 24 inches in size, on a scale showing not more than 200 feet 
to an inch. 

The plat must have upon it a full, detailed written description of the 
land embraced in it, showing the township, range, section and subdi¬ 
vision of section of the land platted. If the premises platted are not 
included in the legal subdivisions of the government survey, then the 
boundaries are to be defined by metes and bounds and courses. 

The plat must contain the full name of the town, city, village or ad¬ 
dition platted; the names of the proprietors and of the person making 
the plat, and the date. 

It must be signed by the proprietors and by the person making it, 
and be witnessed and acknowledged in the same manner as deeds. 

The sections and parts of sections must also be designated on the 
plat by lines with appropriate letters and figures. 

There must be a plain designation of the cardinal points of the com¬ 
pass and a correct scale. 

When complete and before any copies are made from it, the plat 
must be submitted to the Auditor General for his approval, 

lO . The Record.— An exact duplicate of the original plat must 
be filed in the office of the Register of Deeds for the county in which 
the land is situated. Tt must contain all the matter in the original 
plat and the certificate of the Register of Deeds and the person who 
made the original plat, that they have separately carefully compared 
the duplicate with the original plat, and that it is an exact duplicate 
thereof and of the whole of such plat, 


SUBDIVISION OF SECTIONS. 


241 


A third copy must be filed in the office of the Auditor General. 
This copy must contain the certificate of the Register of Deeds and of 
the person who made the plat, that they have separately compared it 
with the duplicate plat on record, and that it is a true transcript there¬ 
from and of the whole of such duplicate plat so recorded. 

The Register of Deeds receives a fee of $2.00 for recording the plat, 
and the sum of $1.00 must accompany the plat filed in the Auditor 
General’s office. 

The law was amended in 1887 so as to require the surveyor to 
plant permanent monuments at all angles in the boundaries of 
the land platted, and at all the intersections of streets, or streets 
and alleys, as shown on the plat; and when there are permanent 
objects in the vicinity of such monuments, the bearings and dis¬ 
tances of such objects to be noted. The character of the monu¬ 
ments and the bearings and distances of such objects or witness 
points must be given in the most convenient manner on the plat. 
The surveyor must certify that the plat is a correct one, and that 
the monuments described in it have been planted as therein 
described. The new provisions of the law are very important. 
The monuments are the crowning work of the survey, without 
which all else is of little value. They mark out the standard of 
measure on the grouud, to which all subsequent surveys must 
conform. President Steele, of the Michigan Engineering Society, 
says: “ Place more monuments instead of less. Place them 

everywhere, no matter whether at the intersections of streets at 
the side lines, the centre lines, or any other lines. Put down all 
you can. Plant them in exact relation one to another. Put the 
bearing on every line, the angle at every intersection. Put it all 
on your plat, and the more you have the better. Leave nothing 
to guess at. Have it so plain that a man who never knew any¬ 
thing about the ground can go there and find all the points.” 


17 


212 A MANUAL OF LAND SURVEYING. 

II. Monuments. —It is more important to a man 
to know precisely where his boundary lines are and that 
they are unchangeable without his consent, than it is that 
he shall have the precise quantity of land ; hence one of 
the most important duties the surveyor has to perform, is 
to fix the most permanent and unmistakable monuments 
to define and preserve boundary lines. This is equally 
true of all original surveys, whether in country or town. 
Mathematical accuracy in measuring distances or running 
lines, fails of its purpose unless there be some means of 
securing an unvarying starting point; while if the land¬ 
marks of the original survey, in accordance with which 
the land was conveyed, be preserved intact, no measure¬ 
ments, good or bad, are needed to define the boundaries. 

Monuments for landmarks should be durable and easily 
distinguishable from other objects in the vicinity. 

They should be accessible, not liable to be moved, and 
their position located by bearings and distances to the 
most permanent objects in the vicinity. 

Various things are used for landmarks—according to 
the nature of the soil and the materials at hand ; chiefly 
wood, stone, earthenware, or iron, in some of their forms. 

Wood. A wooden post, if of suitable size and kind and 
properly planted, makes an excellent landmark, where 
very precise definition of the boundary is not required. 
It should be from 2% to 4 inches in diameter, sound and 
straight and planted vertically in the ground to a depth 
of at least three feet, for permanent purposes. Red cedar 
black-walnut, cherry or white oak hearts make very dur¬ 
able posts. When the post has decayed the rotten wood 
and cavity in the earth preserve the point better than the 
sound post, as they cannot be pulled up nor moved from 
place without moving the surrounding earth with them. 

Stone. If a rough stone or boulder is used for a monu¬ 
ment, it should either be so large as not to be moved by 
any ordinary accident or so firmly imbedded in the earth 
as to defy the plow or the road maker. If of a kind com¬ 
mon in the vicinity, it should be very plainly marked and 
have some foreign material like brick, iron, glass, or 
crockery imbedded around it, to identify it by. 


SUBDIVISIONS OF SECTIONS. 


243 


If cut stone is used, it should be of the best quality and 
be long enough and set deeply enough to insure perma¬ 
nency. If the stone is a soft one it should be protected 
from injury. A stone 36x8x8 inches dressed down at the 
top to 6x6 inches is the size in use in many of the large 
cities for landmarks. It is common to cut a cross or drill 
a hole in the top of the stone to indicate more precisely 
the corner or line. If still greater precision is required a 
piece of metal is set in the stone and the point indicated 
by lines cut as finely as desirable. 

Iron. Monuments of cast iron have been used and are 
excellent. A hollow cone 18 to 36 inches in length with a 
broad flange at the bottom, when set in the ground holds 
its position very firmly and will last indefinitely. Iron 
rods, and pieces of gas pipe are also used. They need to 
be well packed about the top with brick or stone to keep 
them in position. 

Other Materials. Some monuments are made of the 
same material as the earthenware sewer pipe, and burned 
and glazed in a similar manner. They are solid, cylindri¬ 
cal, three inches in diameter and of various lengths. The 
ends are suitably marked before burning. They are 
very convenient to use and durable, but need to be well 
protected. Brick set on end two and two to a depth 
of three feet and packed about the top to prevent 
moving make an excellent monument. Another excel¬ 
lent device is to make a deep hole in the earth, one or two 
inches in diameter, and fill it with a paste of quick lime, 
plaster of paris, or portland cement. 

Protection. A good plan for protecting monuments in 
the streets of a town, is to place them in shallow pits a 
foot or more in diameter. Set the monument in the pit 
so that the top of it shall be several inches above the bot¬ 
tom of the pit and as much below the street pavement. 
Protect it with a cast iron cylinder set about it, having a 
slightly conical cover which is level with and forms part 
of the pavement. The summit of the cover answers some 
of the purposes of the monument, while by removing it 
the monument Itself is brought to view. 


A MANUAL OF LAND SURVEYING. 


244 


CHAPTER X. 

RESURVEYS. 

1. In an old settled country, the principal work of 
the surveyor is to retrace old boundary lines, find old cor¬ 
ners, and relocate them when lost. In performing- this 
duty, he exercises, to a certain extent, judicial functions. 
He usually takes the place of both judge and jury, and 
acting- as arbiter between adjoining proprietors, decides 
both the law and the facts in regard to their boundary 
lines. He does this not because of any right or authority 
he may possess, but because the interested parties volun¬ 
tarily submit their differences to him as an expert in such 
matters, preferring to abide by his decision rather than 
go to law about it. • 

In making- resurveys the surveyor is called upon— 

1. To construe descriptions in deeds; 

2. To find the location of corners and boundary lines; 

3. To renew corner monuments and to mark anew 
boundary lines. 

2. In construing the descriptions the following 

rules have been laid down by the courts: 

Rule 1. The description of boundaries in a deed is to 
be taken most strongly against the grantor. 

Marshall v. Niles, 8 Conn. 369. 

Ryan v. Wilson, 9 Mich. 262. 

2. A deed must be construed according to the condition 
of things at the date thereof. 

Crogan v. Burling Mills, 124 Mass, 390. 

Written descriptions of property are to be interpreted 


RESURVEYS. 245 

in the light of the facts known to and in the minds of 
the parties at the time. 

Wiley v. Sanders, 36 Mich. 60. 

McConnell v. Rathbun, 46 Mich. 305. 

And should be construed with reference to any plats, 
facts, and monuments on the ground referred to in the 
instrument. 

Anderson v. Baughman, 7 Mich. 77. 

Bowen v. Earl, 28 Mich. 538. 

3. Where the description of the boundaries in a deed 
are indefinite or uncertain, the construction given by the 
parties, and manifested by their acts on the ground, is 
deemed the true one unless the contrary is clearly shown 

Reed v. Prop. Locks and Canals, 8 How. (U. S.) 274. 

4. Every call in the description of the premises in a 
deed must be answered if it can be done, and none is to 
be rejected if all the parts can stand consistently together. 

Herrick v. Hopkins, 10 Shep. (Me.) 217. 

5. Where the boundaries mentioned are inconsistent 
with each other, those are to be retained which best sub¬ 
serve the prevailing intention manifested on the face of 
the deed. 

Gates v. Lewis, 7 Vt. 511. 

6. The certain description must prevail over the uncer¬ 
tain, in absence of controlling circumstances. 

Richer v. Barry, 34 Me., 116. 

Tewksbury v. French, 44 Mich. 102. 

See also 35 N. H. 121, and 11 Conn. 335. 

7. When one part of the description in a deed is false 
and impossible, but by rejecting that a perfect descrip¬ 
tion remains, such false and impossible part should be re¬ 
jected and the deed held good. 

Anderson v. Baughman, 7 Mich. 79. 

Johnston v Scott, ll Mich. 232. 

8. A deed is to be construed so as to make it effectual 
rather than void. {Ibid.) 

9. Where the description in a deed calls for land “ owned 
and occupied,” the actual line of occupation is a material 


246 


A MANUAL OF LAND SURVEYING. 


call to be considered in locating* the lines of the land 

bounded therein. 

Fahey v. Marsh, 40 Mich. 239. 

Cronin v. Gore, 38 Mich. 38G. 

10. Where land is described as running* a certain dis¬ 
tance by measure to a known line, that line will control 
the measure and determine the extent of the grant. 

Flagg v. Thurston, 13 Pick. (N. Y.) 145. 

See also 13 Wend. (N. Y.) 300, and 7 Iredell. (N. C.) 169 and 310. 

11. Not so if the line is obscure, not definitely fixed, 
marked or known, and therefore likely to be looked upon 
by the parties as less certain than the measure given. 

Howell v. Merrill, 30 Mich. 282. 

12. In the case of Land Co. v. Saunders in 5th Otto 
(U. S.), the Supreme Court of the United States held the 
west line of Hart’s location to be the boundary of a 
grant. It was in a mountainous country and had never 
been surveyed or marked—although capable of being 
marked—the line being simply marked on the plat of the 
location. This line is held to be such a monument as 
would control course and distance. 

13. Where land is conveyed “beginning at” and bound¬ 
ing land of “B,” the point of beginning and boundary 
is the true line of B’s land, and not the line of occupa¬ 
tion as shown by a fence set up and maintained by B be¬ 
fore and after the conveyance, with the consent of the 
owner of the lot conveyed, under the mistaken belief 
that such was the true line. 

Cleveland v. Flagg. 4 Cusliing (Mass.) 76. 

14. A course from corner to corner means prima facie 
a right line; but this may be explained, by other matters 
in the case, to be a crooked or curved line, as following a 
ditch, or hedge, or stream. 

Baker v. Talbott, 6 Mont. (Ky.) 182. 

15. “Northward” or “northerly” means due north 
when nothing is mentioned to show the deflection of the 
course to the east or west. 

Jackson v. Reeves, 3 Caines, N. Y. 293. 

Brandt v, Ogden, 1 Johns, N, Y. 156. 


RESURVEYS. 


247 


16. The use of the term “ about ” indicates that exact 
precision is not intended; but where nothing more cer¬ 
tain can be found to control the course and distance, the 
grantee is limited to the exact course and distance given. 

Cutts v. King, 3 Greenl. Me. 482.1 

17. Where a given quantity of land is to be laid off on 
a given base, it must be included in four lines, so that the 
lines proceeding from the base shall be at right angles 
with it, and the line opposite the base shall be parallel 
with it, unless this form is repugnant to the entry. 

Massie v. Watts, 6 Cranch. (U. S.) 148. 

Ker v. Watts, 6 Wheat. (U. S.) 550. 

18. Seventy acres lying and being in the southwest cor¬ 
ner of a section, is a good description, and the land will 
be in a square. 

Walsh v. Ringer. 2 Ham. (Ohio) 327. 

19. Where lines are laid down on a map or plan, and 
are referred to in a conveyance of land, the courses, dis¬ 
tances, and other particulars appearing on such plan are 
to be as much considered the true description of the land 
conveyed as they would if expressly recited in the deed. 

Davis v. Rainsford, 17 Mass. 211. 

See also 14 Mass. 149, and 1st Greenl. Me. 219. 

20. A conveyance by metes and bounds will carry all 
the land included within them, although it be more or 
less than is stated in the deed. 

Butler v. Widger, 7 Cow. (N. Y.) 723. 

Bratton v. Clawson, 3 Strobli. S. C. 127. 

Gillman v. Riopelle, 18 Mich. 164. 

21. A grant of land bounded by a highway takes to the 
center of the highway. If it be designed to exclude the 
highway, it must be so stated in explicit terms. 

Champlin v. Pendleton, 13 Conn. 23. 

See also 7 N. H. 275; 6 Shep. Me. 276. 

Furkiss v. Benson, 28 Mich. 538. 

A deed of land lying east of a certain street, and ex¬ 
plicitly bounded by the east line of the street, conveys no 
title to the soil in the street. 

G. R. & I. R. R. Co. v. Mary Heisel, 38 Mich. 62. 


248 


A MANUAL OF LAND SURVEYING. 


22. Tlie mention of quantity of acres after a definite 
description by metes and bounds, or by the aliquot part of 
the section, is a matter of description only, and quantity 
being the least certain, does not control. 

Amich v. Holman, 13 Strobli. S. C. 132. 

McClintock v. Rogers, 11 Ills. 279. 

Martin v. Carlin, 19 Wis. 454. 

23. Where boundaries are doubtful, then quantity often 
becomes a controlling consideration. 

Winans v. Cheney, 55 Cal. 567. 

24. Grants by government are to be construed accord¬ 
ing to the common law, unless it has done some act to 
exclude that construction. 

m 

Middleton v. Pritchard, 3 Scam. Ills. 510. 

The references in the following recent decisions are to 
the several “Law Reporters,” published by the West 
Publishing Co., of St. Paul, Minn. 

25. A reference in a description to the government 
patent, makes the patent description and the government 
survey a part of the deed. 

Miller v. Topeka Land Co., (Kan.) 24 P. 420. 

26. Where a survey is referred to in a deed for greater 
certainty, it legally forms a part of it and both should be 
construed together. 

Heffleman v. Otsego Water-Power Co., (Mich.) 43 N. W. 1096. 

27. Extrinsic evidence is always admissible to explain 
the calls of a deed for the purpose of applying them to 
the subject-matter, and thus to give effect to the deed. 

Thompson v. Southern Cal. M. R. Co., (Cal.) 23 P. 130. 

28. An exception in a deed which reads, “ Except the 
dower of fifty acres, as fully described in the deed given 
the C. B. Co.,” is not void, though the boundaries of the 
excepted land are not defined in any way, as reference 
may be had to the deed to the C. B. Co. to ascertain them. 

McAffee v. Arline, (Ga.) 10 S. E. 441. 




REStmVEYS. 


m 


29. A deed conveying property by lot numbers is not 
void for uncertainty, though the recorded plat shows no 
division of the blocks into lots; it being shown that the 
proprietors had always treated the blocks as divided 
into lots, and that for many years the property had been 
assessed, conveyed, and generally known by the lot num¬ 
bers. 

Marvin v. Elliott, (Mo.) 12 S. W. 899. 

30. A deed describing the granted premises as “sub¬ 
division of lot Xo. 4 of division No. 1G,” etc., followed by 
the total number of acres contained in lot 4, and then 
excepting land previously sold, is not void for indefinite¬ 
ness, though lot 4 was never subdivided, as it evinces a 
clear intent to convey the balance of whatever land the 
grantor owned in lot 4; and the deed will be construed as 
though the word “of” after the word “subdivision” had 
been omitted. 

Weeks v. Martin, 10 N. Y. S. 656. 

31. A deed to a railroad company of a right of way 
“along the line as surveyed and laid out” by the com¬ 
pany’s engineer is not void for uncertainty where it ap¬ 
pears that when the deed was executed the line of the 
road had been surveyed and distinctly marked by stakes 
stuck in the ground, and that subsequently the road was 
constructed following the exact line of the survey. 

Thompson v. Southern Cal. M. R. Co., (Cal.) 23 P. 130. 

32. In a deed of land by metes and bounds, an exception, 
of “lot G, block 3G, heretofore conveyed to 11,” excepts 
a lot so numbered on a plat made by the grantor and 
grantee, but not then recorded, there being no other lot 6 
block 3G, within the land granted. The recital of a con¬ 
veyance to 11. may be rejected as a falsa demonstratio. 

Ambs v. Chicago, St. P., M. & O. R’y Co., (Minn.) 46 N. W. 321. 

33. Though a plat be incomplete as respects the loca¬ 
tion of monuments, or in respect to measurements and 
distances, yet where land so surveyed has been conveyed 


250 


A MANUAL OF LAND SURVEYING 


by reference thereto, and the location of the lots so 
conveyed and designated is well known by all parties 
interested, and susceptible of identification according 
to the actual survey on the ground, the description is 
sufficient to pass the title. 

Bolirer v. Lange (Minn.) 4G N. W. 358. 

34. The description in a deed was : “ Beginning at * 

* * ; running thence northeasterly, along Grove street, 

25 feet; and thence northwesterly, and parallel with 
Woodruff avenue, 108 feet 9 inches, to lot No. 80, on said 
map; thence southwesterly, along lot No. 80, 25 feet; and 
thence southeasterly, and parallel with Woodruff avenue, 
108 feet 9 in., to the westerly side of Grove street, the 
point or place of beginning.” Lines drawn from Grove 
street, 108 feet 9 inches, parallel to Woodruff avenue, 
would not reach lot 80 by 5 inches. Held , that there was 
a mistake in describing the length of the lines parallel to 
Woodruff avenue, and that it was intended that they 
should extend 109 feet 2 inches, and not that they should 
run in such a direction that they would reach lot 80 at 
the distance of 108 feet 9 inches from Grove street. 

Casey v. Dunn, 8 N. Y. S. 305. 

35. It being stated with certainty in the deed that such 
lines were parallel to Woodruff avenue, it is immaterial, 
in construing the description, that the corresponding 
lines in the conveyances of neighboring property were 
at right angles to Grove street, instead of being parallel 
to Woodruff avenue. 

Casey v. Dunn, 8 N. Y. S. 305. 


36. The description in a deed was certain as to the 
northern and western boundaries. The course of the 
eastern boundary was south for a distance of 8 rods. The 
southern boundary was “ then west, in a line parallel to, 
and eight rods south of,” the northern boundary, “one 
hundred and sixty-two feet, to” the western boundary. 
By reference to another deed, it was made certain that 
the north 6 rods of the eastern boundary was a straight 
wall. The course of the other 2 rods was uncertain. 


RESURVEYS. 


251 


Extending the line of the wall 2 rods south, and from the 
end of this line drawing a line parallel to the northern 
boundary, to the western boundary, a southern boundary 
165 feet in length would be obtained. Held , that from 
the southern end of the wall the eastern line should be 
deflected towards the west at such an angle that at the 
distance of 2 rods it would intersect a line parallel with 
the northern boundary at the distance of 162 feet from 
the western boundary. 

Ladies’ Seamen’s Friend Soc. v. Halstead, (Conn.) 19 A. G58. 

37. A city, by its president and trustees, conveyed to 
defendants’ grantor “that lot of land containing 60 acres, 
lying in block No. 1111, according to the official map of 
said city made by * * * A. D. 1856.” The deed re¬ 
ferred to a resolution of the trustees, under which the 
lands were sold, which provided that all surveys should 
be made by the purchaser. At the time of the deed there 
had been no survey or subdivision of the block. Held , 
that the deed conveyed an undivided 60 acres of the block. 

Cullen v. Sprigg,<Cal.) 23 P. 222. 

38. Where a description by metes and bounds is supple¬ 
mented by a reference to a particular subdivision of land 
to indicate the tract intended to be conveyed, the former 
will not necessarily be controlling, when it would leave a 
strip 13 feet front by 100 deep in the grantor, which 
clearly appears to have been intended to be conveyed by 
the latter description. 

Cannon v. Emmons, (Minn.) 4G N. W. 35G. 

39. Ordinarily, calls for natural or artificial monuments 
will control courses and distances; but a call for course 
and distance will not be subordinated to a call for an un¬ 
marked line in a prairie, which cannot itself be ascertained 
except by running the boundaries of another survey 
according to course and distance. 

Johnson v. Archibald, (Tex.) 14 S. W. 26G. 



252 A MANUAL OP LAND SURVEYING. 

40. A complaint was filed to quiet title to 150 acres 
of land lying on the south side of a fractional section. A 
surveyor was ordered to survey that quantity, to be taken 
the full length of the section from the east side thereof 
to a river as the western boundary, and extending far 
enough north to include 150 acres. The surveyor executed 
the order, and reported a survey, which was accepted, and 
the court entered judgment, wherein the land was doubly 
described by inconsistent descriptions. The first des¬ 
cribed it as in the order of survey, and the second by 
metes and bounds, by which, after beginning at the 
southeast corner of the section, and following the south 
line to the river, it ran up the river, with tho meanders 
thereof, to a stake placed by said surveyor 19^ chains 
north of the south line of the section; thence running 
westerly, parallel with the south line, 53.04 chains, to a 
stake in the east line of the section; and thence southerly 
with said line 9} chains, to the beginning. The stakes 
were gone, but were shown to have been placed at points 
19i chains from the south line, thereby including 150 
acres. Held , that the first description should govern. 

Caspar v. Jamison, (Ind.) 21 N. E. 743. 

41. Under a deed of land bounded by a street, according 
to a map referred to, the line of the street as actually sur¬ 
veyed is the boundary of the land conveyed. 

Andreu v. Watkins, (Fla.) 7 So. S7G. 

42. A deed described the land conveyed as “commenc¬ 
ing oil the 8. road at the north-east corner of the land 
owned by S.; running south, to the south-east corner of 
said S.’s land, two acres; from thence, easterly and 
parallel with said S. road, two acres; thence running 
northerly two acres, until it strikes said road; and thence 
westerly, along said road, two acres, to the beginning; 
containing four acres of land, neither more nor less.” 
Held , that as the description by quantity so clearly shows 
the intention to limit the grant to four acres in rec¬ 
tangular form, and as the length of the west line is given, 
the intention must control distances. 

Rioux v. Cormier, (Wis.) 44 N. W. G54. 


RESURVEYS. 


253 


i A similar construction is to be given the United States 
\. statute providing for the survey in certain cases of tracts 
i of land two acres in width and running back a depth of 
f forty acres. B. S. 2407. 

43. A city condemned a strip of land for railroad and 
sewer purposes, and, after constructing a road-bed along 
this, it conveyed to a railroad company “its title to the 
road bed, bridges, and right of way” along the entire 
route, and “ all the land belonging to the city,” between 
certain streets, “for depot purposes.” The company had 
formerly occupied a right of way for a double track on 
other streets, and the city, in consideration of the change 
of the railway to the street forming the line of the road 

i in the conveyance, agreed to furnish the company a road¬ 
bed. Held, that outside of the part conveyed for depot 
purposes nothing but the road-bed was conveyed. 

Long v. Louisville & N. R. Co., (Ivy.) 13 S. W. 3. 

44. The deed of a city lot, and plat with reference to 
which it was made, called for the south line of Cherry 
street as the northern boundary of the lot. The line re¬ 
ferred to had been established by the City Surveyor 37 
years before and ever since acquiesced in. The other lots 
in the block had been bought, fenced and built upon on 
the assumption that this survey was correct. A more 
recent survey tended to show that the line was three feet 
too far north. Held, that the presumption of correctness 
was with the older survey, and as the lot owner had got 
all he bargained for, and the later survey would cause the 
lines of the other lots to cut into the buildings, the older 
survey must prevail. 

Wilmarth v. Woodcock, Mich. 33 N. W. Rep. 401. 

45. A description in a deed reads: The east % of the 
east 34 of the northwest and the east 34 of the east 34 
of the southwest frac. 3s£,” etc., “containing 50 acres of 
land; being the east half of 100 acres conveyed by A. 
and B. to E. The south line of the tract is irregular on a 
lake, and a line north and south through the center of the 
tract would give one parcel nine acres more land than 
the other. Held, that the language is apt and proper to 





254 A MANUAL OF LAND SURVEYING. 

divide the tract by a north and south line which would 
give to each 50 acres, or one-half of the whole. 

A description of the half of the parcel of land, accord¬ 
ing to the United States survey, would have excluded the 
idea of equal quantities and fixed the dividing line in 
accordance with the Act of Congress. If any other line 
had been agreed upon between the owners as the bound¬ 
ary line, it would govern the case. 

Jones v. Pashby, Mich. 29 N. W. Rep. 376. 

Dart v. Barbour, 32 Mich. 276. 

Heyer v. Lee, 40 Mich. 353. 

46. A description in a deed, if otherwise good, is not 
vitiated by the omission of the word “rods” to avoid 
tautology, when the meaning is plain. 

Taber v. Shattuck, 55 Mich. 370. 


1. Adverse Possession. —When the boundary line 
between the lands owned by adjoining land-owners is 
unknown, they may by parol fix a line between each 
party, each party mutually agreeing thereto and acting 
thereon, which is binding between them; but if the line 
is known, then the transfer of any portion of the land on 
one side of the line from the one to the other must be in 
writing, to be valid. 

Jinkins v. Trager, 40 F. 726. 

2. The adverse possession of land by a grantor cannot 
avail his grantee, beyond the boundary line described in 
the deed. 

Jenkins v. Trager, 40 F. 726. 


3. Possession as owner is an essential condition by 
which the ownership of immovables can be acquired 
without title, or possession in good faith. 

Stille v. Sclmll, (La.) 6 So. 634. 

4. Continuous possession of land for more than 30 
years under claim of ownership, though without color of 
title, constitutes title in fee. 

Bowen v. Swander, (Ind.) 22 N. E. 725, 


RESURVEYS. 


255 


5. One cannot acquire title to land by adverse posses¬ 
sion where he claims title under a deed which in fact 
does not include such land in its description. 

Casey v. Dunn, 8 N. Y. S. 305. 

6. Where title is claimed by adverse possession, if the 
possession is by actual occupation of the possessor under 
claim of*title, it is visible, open, notorious, distinct, and 
will be presumed to be hostile. 

Green v. Anglemire, (Mich.) 43 N. W. 772. 

7. Where the line between adjoining owners is in doubt, 
but they only claim ownership to the true line, wherever 
that may be, no title by adverse possession can arise in 
either, as against the other. 

Krider v. Milner, (Mo.) 12 S. W. 4<;i. 

3. In construing deeds conveying title to lands 
bordering on waters, it will be necessary for the 
surveyor to inquire into the local laws of the State in 
which the premises lie, as different States by their laws 
and courts give different constructions to the word 
“navigable ” as applied to streams and the smaller lakes. 
The statute of the United States provides that 

“ Sec. 440. All navigable rivers, within the territory occupied by the 
public lands, shall remain and be deemed public highways; and, in all 
cases where the opposite banks of any streams not navigable belong 
to different persons, the stream and the bed thereof shall become 
common to both.” (1 Stat. 468 ; 2 id. 235; R. S. 2476.) 

It is a universal rule that grants of land bordering on 
navigable streams take only to high-water mark, while 
grants on non-navigable streams take to the center of the 
stream, or tXwfilum aqua’, as it is termed. 

Now, whether the proprietor in any given case owns 
the land under water to the center of the stream, or only 
takes to high-water mark, depends on the local construc¬ 
tion given to this word navigable. 

Under the Common Law , a navigable stream is one in 
which the tide ebbs and flows. Some exceptions to the 
rule are made in England. 





256 


A MANUAL OF LAND SURVEYING. 


Under the Civil Law , a navigable stream is one capa¬ 
ble of being used as a highway of commerce. In the 
case of the Railroad Co. v. Schurmier, (7 Wallace, 272), 
the Supreme Court of the United States says that “the 
words navigable and non-navigable were applied by Con¬ 
gress without respect to the ebb and flow of the tide,” 
and in the case of Bowman and Burnley v. Wathieu and 
others. (2d McLean, 276), they say that “ the common law 
doctrine as to the navigableness of streams can have no 
application in this country, and the fact of navigableness 
does in no respect depend on the ebb and flow of the 
tide.” 

The courts of Pennsylvania, North Carolina, South 
Carolina and Alabama hold the same view. On the con¬ 
trary, in Maine, New Hampshire, Massachusetts, Connec¬ 
ticut, New York, Maryland, Virginia, Ohio, Illinois, In¬ 
diana, and Michigan, the common law doctrine is held to 
prevail. (See Angell on Tide Waters, pp. 77 and 78.) 

Hence, in applying the principles laid down by the 
courts in the following decisions, the surveyor will bear 
in mind the locality in which they are to be applied. 

1. Proprietors of lands bordering on navigable rivers, 
under titles derived from the United States, hold only to 
the stream, as the express provision is, that all such rivers 
shall be deemed to be and remain public highways. 

R. R. Co. v. Schurmeir, 7 Wallace (U. S.) 272. 

2. Where a sea or bay is named as a boundary, the line 
of ordinary high-water mark is always the line, where 
the common law prevails. 

U. S. v. Pacheco, 2 Wallace (U. S.) 587. 

3. A boundary on a stream or by or to a stream in¬ 
cludes flats at least to low-water mark, and in many cases 
to the middle thread of the river. 

Thomas v. Hatch, 3 Sumner (U. S.) 170. 

4. A boundary on the bank of a river referring to fixed 
monuments on the bank, limits the grant to the bank and 
excludes the flats. (Ibid.) 

See also Hopkins v. Kent, 9 Ohio, 13. 


RESURVEYS. 


257 


5 . The words “along the hank” are strong and defi¬ 
nite enough to exclude the idea that any part of the river 
or its bed was granted in the navigable or unnavigable 
parts of the river. 

Howard v. Ingersoll, 13 How. (U. S.) 341, 416. 

A deed describing the land by a boundary running to a 
stream, and thence along its bank, and reserving the right 
to use the river front a specified time, conveys the land 
to the water’s edge and covers the riparian rights to the 
middle of the stream. 

Cole v. Wells, 49 Mich. 450. 

6 . Congress, in making a distinction between streams 
navigable and those not so, in the acts relating to the sale 
of the public lands bordering thereon, intended to pro¬ 
vide that the common law rules of riparian ownership 
should apply to the lands bordering on the latter, but that 
the title to lands bordering on the former should stop at 
the stream. 

R. R. Co. v. Scliurmeir, 7 Wall. (U. S.) 272. 

7. In streams which are not navigable, adjacent pro¬ 
prietors own to the center of the stream measured from 
low-water mark. 

Clark v. Caupau, 19 Midi. 325. 

Moore v. Sanborn, 2 Mich. 519. 

Lorman v. Benson, 8 Mich. 18. 

Bay City Gas Light Co. v. Jnd. Wks 8 Mich. 182. 

Lamb v. Ricketts, 11 Ohio, 311. 

8 . The same principle is applied to Lake Muskegon,in 
Michigan, (Rice v. Ruddeman, 10 Mich. 125), but not ap¬ 
plied to a similar lake in Wisconsin, where the court says, 
(Deidrich v. N. W. U. Ry. Co., 42 Wis. 271): “Riparian 
owners upon a natural lake or pond take only to the 
shore.” 

In the case of the State of Indiana v. Milk, Circuit 
Court of the United States, April term, 1882,11th Bissell^ 
page 197, the court rejects the theory of riparian owner¬ 
ship in the lake, and after presenting its reasons at some 
18 


258 


A MANUAL OF LAND SURVEYING. 


length, concludes with the following: “ That while a gen¬ 
eral grant of land on a river or stream non-navigable 
extends the line of the grantee to the middle or thread of 
the current, a grant on a natural pond or lake extends 
only to the water’s edge.” 

9 . Islands in rivers fall under the same rule as to own- 
nership as the soil under water does. If not otherwise 
lawfully appropriated, they belong to the proprietors on 
either side of the stream, according to the original divid¬ 
ing line or filum aquce as it would run if the islands were 
under water. The fihnn aquce is midway between the 
lines of ordinary low-water mark, without regard to the 
channel or depth of water. When t lie island is appropri¬ 
ated, the boundary is then midway between it and the 
mainland. 

McCullough v. Wall, 4 Rich. (S, C.) 68. 

Kimball v. Schaff, 40 N. H. 190. 

10 . The grant includes any land between the meander 
line and the water, in an unnavigable stream. 

The same principle applies to unnavigable lakes. 

Forsyth v. Smale, 7 Biss. (U. S.) 201. 

11 . High-water mark in the Mississippi River is to be 
determined from the river bed, and that only is river bed 
which the river occupies long enough to wrest it from 
vegetation. 

Houghton v. Railway Co., 47 Iowa, 370. 

12. A bank is the continuous margin where vegetation 
ceases. The shore is the sandy space between it and low- 
water mark. 

McCullough v. Wainwright, 14 Penn. St. 59. 

13. Where a levee was shown to have been judiciously 
located by a competent engineer and agents of the State 
acting under authority conferred by the State Legislature, 
it was held that such levee became the boundary line of 
high water, and that no private ownership could be ac¬ 
quired to land lying between that and the bed of the 
stream. 

Musser v. Hershey, 42 Iowa, 356. 


RESURVEYS. 


259 

14. Grant of a city lot bounded on a river, takes to the 
center of the stream. 

Watson v. Peters, 26 Mich. 50S. 

Riparian rights, unless expressly limited, extend to the 
middle of the navigable channel, and cover any shallows 
or middle ground not shown in the government surveys, 
but lying between such shallows and the shore, and it 
makes no difference that the deed conveying the premises 
to which the rights attach describes them according to a 
city plat instead of the government entry. 

Fletcher v. Thunder Bay Boom Co., 51 Mich. 277. 

15. But if the plat plainly indicates the proprietor’s 
ntent to reserve the space between the shore and the 
thread or main channel, the case would be different. 

Watson v. Peters, 26 Mich. 50S. 

16. Riparian rights extend laterally into the streem. 
Rocks and shoals along the margin of navigable rivers 
above tide-water belong to the riparian owner. 

Moore v. Willamette T. and L. Co., 7 Oregon R. 355. 

17. When a navigable stream is meandered in making 
the public surveys, and the United States has granted to 
the meander line, the grantee takes to the river. The 
stream, and not the meander line, is the true boundary of 
the riparian owner. 

Minto v. Delaney, id., 337. 

18. Lands patented by the United States on a tide-water 
stream extend to the meandered line of the stream, which 
is the line of ordinary high water. 

Parker v. Taylor, id., 435. 

19. A boundary by the shore of a mill pond takes to 

low water mark. 

Stevens v. King, 76 Maine 197. 

20. N. conveyed a lot according to a certain plat. The 
plat represented the lot as bounded north by a street 
south by a stream; on the east and west by lines running 
from the street to the stream, with figures purporting to 
give the length of these lines. In fact, the distance to 
the stream was greater than indicated by these figures. 


260 


4 MANUAL OF LAND SURVEYING. 


Held, that the conveyance of the lot according to the plat 
included all the land between the street and the stream. 

Nicolin v. Schneiderham, Minn. 33, N. W Rep. 33. 

In Turner vs. Holland, the Supreme Court of Michigan 
gives riparian rights to owners of lots bounded by a bayou 
of Saginaw river, described by plat similar to the above. 

33 N. W. Rep. 233. 

21. In a navigable stream, as the DesMoines river in 
Iowa, high water mark is the boundary line. When, by 
action of the water, the river bed changes, high water 
mark changes and ownership of adjoining land changes 
with it. The location of meander lines does not affect 
the question. Meander lines are not boundary lines. 

Steele v. Sanchez, 33 N. W. Rep. 3G7. 

Krant v. Crawford, 10 Iowa 349. 

Lockwood v. R. R. Co., 37 Conn. 387. 

22. A boundary stated in a deed as a line forty feet 
above the border of a river at high water mark, is not 
ambiguous, and if disputed is to be fixed like any other 
fact, by testimony and an examination of the ground. 

Bresler v. l’itts, 59 Mich. 348. 

Recent decisions from “Law Reporters:” 

23. A patent for a fractional quarter section, which 
is bounded by a meandered stream, passes title to all 
land within the lines of said quarter section between 
the meandered line and the water’s edge. 

Splmng v. Moore, (Ind.) 22 N. E. 319. 

24. The owner of land on the margin of a navigable 
stream, holding under a grant from the United States, 
does not take to the middle of the stream, but to high- 
water mark, which is determined by the change in the 
vegetation and the character of the soil, and the beds of 
all navigable streams, though the tide does not ebb and 
flow in them, belong to the state. 

St. Louis, I. M. & S. Ry. Co. v. Ramsey, (Ark.) 13 S. W. 931. 

25. The owner of land on a bay conveyed an acre at 

the end of the tract nearest the bay, described as follows: 
“.Beginning * * * by the beach, running * * * 


RESURVEYS. 


261 


along- the beach to,” etc. In the general description of 
the tract it was bounded “easterly by the said beach.” 
The grantee was given the privilege of a road from the 
middle of the front of the lot to the bay, and also half 
the drift coming on shore in front of the lot, and all the 
other privileges of the beach were reserved by the grant¬ 
or, who bound himself not to build any house in front of 
the lot. The courses and distances would not carry the 
boundary to high-water mark. Held, that the beach did 
not pass by the deed. 

Benson v. Townsend, 7 N. Y. S. 102. 

26. Where two deeds in plaintiff’s chain of title res¬ 
pectively define the boundary of the land “by the edge of 
the mill-pond” and as “the bank of said mill-pond,” and 
defendant is entitled to pond as much land as the pond 
flowed at the time of his purchase, defendant may enter 
on land originally covered by the pond, but which has 
subsequently become dry land by the receding of the 
water, though plaintiff’s deed on its face shows his line 
to be the center of the pond. 

Holden v. Chandler, (Vt.) IS A. 310. 

27. Where the patentee of “the north half of the south¬ 
east quarter, and that part of the northeast fractional 
quarter, of Section 36,” etc., “which lies north of the 
Kankakee river, containing in all 122.70 acres,” conveys 
“the northeast quarter of Section 36,” etc., “containing 
122.70 acres,” the deed passes title to all of the land in 
said northeast fractional quarter lying south of said river. 

Sphung v. Moore, (Ind.) 22 N. E. 310. 

28. Where one who owns a tract of land that surrounds 
and underlies a non-navigable lake, the length of which 
is distinguishably greater than its breadth, conveys a 
parcel thereof that borders on the lake, by a description 
which makes the lake one of its boundaries, the presump¬ 
tion is that the parties do not intend that the grantor 
should retain the title to the land between the edge of the 
water and the center of the lake, and the title of the 
purchaser, therefore, will extend to the center thereof. 

Lembeck v. Nye, (Ohio) 24 !N. E. G8G. 


/ 


262 


A MANUAL OF LAND SURVEYING. 


29. A patent from the United States of a surveyed 
fractional government subdivision, hounded on a me¬ 
andered lake, conveys the land to the lake, although the 
meander line of the survey be found to be not coincident 
with the shore line. 

Everson v. City of Waseca, (Minn.) 40 N. W. 405. 

30. Where the description is by metes and bounds, no 
reference being made therein to the lake, then only the 
land included within the lines as fixed by the terms used 
by the parties to the deed will pass to the grantee. 

Lembeck v. Nye, (Ohio) 24 N. E. G8G. 

31. If, however, the call in the description be to and 
thence along the margin of the lake, no such presumption 
arises, and the title of the purchaser will extend to low- 
water mark only. 

Lembeck v. Nye, (Ohio) 24 N. E. G86. 

32. Where a deed conveys land “bounded and des¬ 
cribed according to ” a certain survey, does not call for 
a river, but calls for a line run between certain points, 
designated by the surveyor as on the bank of a navigable 
river, and it appears that the lines of such survey exclude 
flats between high and low water marks, evidence ali¬ 
unde is admissible that the bank referred to was an 
artificial dike; that the grantee had notice that the 
grantors reserved the flats; that the grantors refused to 
execute a deed expressly conveying the flats; and that 
the sale was expressly subject to the survey, as tending 
to show that the flats were excluded, whatever may be 
the presumption from the deed. 

Palmer v. Farrell, (Pa.) 18 A. 7G1. 

For further rulings, see Boundary Lines. 

Second. 

4. In locating the corners and boundary 
lines on the ground, we will consider: 

1. General rules which apply to all resurveys; 

2. Special applications of these rules to the rectangular 
system of United States surveys. 


RESURYEYS. 


263 


GENERAL RULES. 

IiULE 1.—In locating- a deed on the ground, we are to 
rely— 

(1) On the actual lines originally surveyed; 

(2) On lines run from acknowledged calls and corners, 

(3) On lines run according to the course and distance 
in the deed. 

Avery v. Baum, Wright’s Ohio, 57G. 

1 Rich. (S. C.) 491. 

2. When the boundaries of lands are fixed, known and 
unquestionable monuments, though neither courses, dis¬ 
tances, nor computed contents correspond, the monu¬ 
ments must govern. 

Pernam v. Wead, G Mass. 131. 

Nelson v. Hall, 1 McLean (U. S.) 518. 

3. Marked lines and corners control courses and dis¬ 
tances. Surplus lands do not vitiate a survey nor does a 
deficiency of acres called for in a survey operate against 
it. Wherever the boundaries can be established, they 
must prevail. 

Robinson v. Moore, 4 McLean (U. S. C. C.) 279. 

Morrow o. Whitney, 5 Otto (U. S.) 551. 

4. A deed called for posts as corners. The survey was 
made and the posts set prior to the execution of the deed. 
It was afterward found that there was a shortage of sev¬ 
eral acres. Held that proof that posts were set up as 
corners between adjoining owners controls the call for 
course and distance. 

Alseire v. Ilulse, 5 Ohio, 534. 

5. The rule that courses, distances and quantities must 
yield to monuments, is not inflexible, especially when the 
distances are very short, and the monuments artificial 
ones, as here, a mill-race, etc. 

Higinbotham v. Stoddard, 72 N. Y. 94. 

Ga. Ii. It. Co. v. Hamilton 59 Ga. 171. 

In a case where no mistake could be reasonably sup¬ 
posed in the courses and distances, the reasons of the rule 
were held to fail, and the rule was not applied. 

Davis v. Rainsford, 17 Mass. 207. 


264 


A MANUAL OF LAND SURVEYING. 


6 . The rule that natural or artificial boundaries will 
control distances or courses, authorizes no other depart¬ 
ure from the course or distance than such as is necessary 
to effectuate the apparent intent of the grantor. 

Distances may be increased and courses departed from 
in order to preserve the boundary, but the rule authorizes 
no other departure from the course and distance than 
such as is necessary to preserve the boundary. 

Johnson v. McMillan, 1 Strobli. (S, C.) 143. 

7. If the courses and distances cannot be otherwise rec¬ 
onciled with the monuments in a description, a line in a 
survey which has evidently been omitted will be sup¬ 
plied to prevent the obvious intent of the grantor from 
being frustrated. 

Serrano v. Rawson, 47 Cal. 52. 

See also Schultz v. Young, 3 Iredell, N. C. 385, 
where two lines must be run instead of the one called for, 

to best conform with the whole description in the deed. 

8 . A survey must be closed in some way or other. If 
this can only be done by following the course the proper 
distance, then it would seem that distance should prevail; 
but when the distance falls short of closing, and the 

course will do it, the reason for observing distance fails, 

Doe v, King, 3 How. Miss. 125. 

9. It is a universal rule that course and distance yield 
to natural and ascertained objects. But where these ob¬ 
jects are wanting, and the course and distance can not be 
reconciled, there is no universal rule that obliges us to 
prefer the one to the other. Cases may exist in which 
either one may be preferred, according to the circum¬ 
stances. 

Preston’s Heirs v. Bowman, 0 Wall. (U. S.) 580. 

10. If no principle of location be violated by closing 
from either of two points, that may be closed from which 
will be more against the grantor and include the greater 
quantity of land. 

Johnson v. McMillan, 1 Strobli, S. C. 143. 


RESURVEYS. 265 

11. The boundary line is to be ascertained by running 
direct lines from one monument to the other. 

Melcher v. Merryman, 4 Me. GOl. 

12. A line actually marked must be adhered to, though 
not a right line from corner to corner. Where a line has 
been marked only part of the way, the remainder of the 
line must run direct to the corner called for. 

Cowan v. Fauntleroy, 2 Bibb (Ky.) 2G1. 

13. A marked line of another tract, when called for in 
a conveyance, must be run disregarding distance; but 
where such line can not be established, the distance run 
must govern. 

Gause v. Perkins, 2 Jones Law Rep. (N. Y.) 222. 

14. Where a line is described as running a certain dis¬ 
tance to a particular monument, and that monument has 
disappeared and its place cannot be ascertained, the course 
and distance, in the absence of other controlling words, 
must govern. 

Rudd v. Brooke, 3 Gdl (S. C ) 198. 

See also, Bruckner v. Lawrence, l Douglass (Mich.) 19. 

15. Course and distance yield to known, visible and defi¬ 
nite objects; but they do not yield unless to calls more 

material and equally certain. 

Shipp et cil. v. Miller’s Heirs, 2 Wheat. (U. S.) 31G. 

Courses and distances in the deed are not to be con¬ 
trolled by monuments or objects variant therefrom and 
not called for in the description, but they must yield to 
such objects and monuments as are referred to. 

Bruckner’s Lessee v. Lawrence, l Doug., Mich., 29. 

Moore v. People, 2 Doug., Mich., 424. 

Bower v. Earle, 18 Mich. 1G5. 

16. Wherever it can be proved that the line was actu¬ 
ally run, was marked, and the corners made, the party 
claiming under the deed will hold accordingly, although 

there is a mistake in the description in the deed. 

Cherry v. Slade, 3 Murph. (N. C.) 82. 

A sold to B lot 7, informing B, at the time of the sale, 
that it was four rods wide, and marking it out upon the 




266 


A MANUAL OF LAND SURVEYING. 


ground. He subsequently sold to C lot 8 and a vacated 
alley one rod in width between lots 7 and 8, informing C, 
at the time, that lot 8 was four rods wide, and the alley 
one rod wide, making five rods in all, and pointing out to 
C the marks previously made by him for the boundary of 
lot 7, sold to B, as being also the boundary of the alley 
sold to 0. The premises were occupied by B and C in 
accordance therewith, without dispute. It was subse¬ 
quently found, by reference to the plat, that lot 7 was five 
rods wide, and that there was no alley between the lots; 
whereupon B claimed the additional rod. Held, that to 
allow B to hold the rod in width of land which she did 
not purchase or pay for, and to deprive C of land which 
he did purchase and pay for, would be both bad law and 
bad morals. 

Bolton v. Eggleston, Iowa. 

N. W. Rep., Vol. 16, p. 62. 

17. Boundary may be proved by any evidence which is 
admissible to establish any other fact. 

Smith v. Prewitt, 2 A. K. Marsh. (Ivy.) 158. 

18. Where no bounds were established, the dividing line 
must be run by aid of the measurements in the deeds, the 
oldest title receiving its full measure first. 

Talbott v. Copeland, 38 Me. 333. 

19. A long established fence is better evidence of actual 
boundaries, settled by practical location, than any survey 
made after the monuments of the original survey have 
disappeared. A resurvey made after the monuments of 
the original survey have disappeared, is for the purpose 
of determining where they were, and not toilere they ought 
to have been. 

Diehl v. Zanger, 39 Mich. 601. 

Hunt’s Lessee v, McHenry and Williams, Wright’s (Ohio) 599. 

20 . Where between the plan and the original survey 
there is a difference in the location of the lines and mon¬ 
uments, the lines and monuments originally marked as 


RESURVEYS. 2G7 

such are to govern, however much they may differ from 
those represented on the plan. 

Ripley v. Barry, 5 Greenl. (Me.) 24. 

See also 2 Greenl. (Me.) 214, and 3 Gr. (Me.) 126. 

21 . But no such rule has obtained where the survey was 
subsequent to the plan. 

Thomas v. Patten, 1 Shep. (Me.) 329. 

22 . Purchasers of town lots have a right to locate them 
according to the stakes which they find planted and rec¬ 
ognized, and no subsequent survey can be allowed to 
unsettle them. The question afterwards is not where 
they should have been, in order to make them correspond 
with the lot lines as they should be if the platting were 
done with mathematical accuracy, but it is whether they 
were planted by authority, and the lots were purchased 
and taken possession of in reliance on them. If such was 
the case, they must govern, notwithstanding any errors 
in locating them. 

Flynn v. Glenny, 51 Mich. 580. 

23. Where two surveys call for each other, there can be 
no vacancy unless the lines marked on the ground con¬ 
tradict the call; and in such case the marked lines must 
govern. 

McGinnis v. Porter, 20 Penn. 80. 

24. Where two surveys made twenty-three years apart 
are found to disagree, the probabilities favor the earlier 
survey when the original corners and witnesses are gone 
at the time of the last survey, especially if the line of 
the first survey has remained unquestioned for many 
years. 

Case v. Trapp, 49 Mich. 61. 

25. When the same grantor conveyed to two persons, to 
each one a lot of land, limiting each to a certain number 
of rods from opposite known bounds running in a direc¬ 
tion to meet if extended far enough, and by measure the 
lots do not join—when it appears from the same deeds 
that it was the intention that they should join, a rule 


268 


A MANUAL OF LAND SURVEYING. 


should be applied which will divide the surplus between 
the grantees in proportion to the length of the respective 
lines as stated in their deeds. 

Lincoln v. Edgecomb, 28 Maine, 275. 

20 . Where original surveys have been made, and re¬ 
turned as a block into the land office, the location of each 
tract therein may be proved by proving the location of 
the block. In ascertaining the location of a tract, the 
inquiry is not where it should or might have been located, 
but where it actually was located. 

Every mark on the ground tending to show the loca¬ 
tion of any tract in the block, is some evidence of the 
location of the whole block, and therefore of each tract 
therein. 

Coal Co. v. Clement, 95 Pa. St. 12G. 

27. Where lots are conveyed by number according to a 
plat which is made from an actual survey, the corners 
and lines fixed by that survey are to be respected. 

Pyke v. Dyke, 2 Greenl., Me., 214. 

28. Streets which are well defined, and designated by 
some natural or artificial monument, must govern course 
and distance in fixing boundaries of lands; but streets 
which are not thus defined, and themselves require to be 
located, would furnish very uncertain guides in arriving 
at the boundaries of other lands. 

Saltenstall v. Riley. 28 Ala. 1G4. 

20 . When streets have been opened and long acquiesced 
in, in supposed conformity to the plat, they should be ac¬ 
cepted as fixed monuments in locating lots or blocks con¬ 
tiguous thereto or fronting thereon. 

Van den Brooks v. Correon, 48 Mich. 283. 

30. Lands have been laid off into lots and blocks, and 
platted, before being cleared, when, by reason of inequal¬ 
ities of the surface, logs, and other obstructions, strictly 
accurate surveys were not and could not be made. Where 
the blocks and streets were staked out at the time, such 


RESURVEYS. 


269 


monuments would be fixed and permanent, leaving the 
excels or shortage to be dealt with by itself. 

So where the streets, although not so designated, have 
by the parties interested or by the public authorities been 
opened, used, and acquiesced in, they thereby become per¬ 
manent boundaries and form new starting points in sub¬ 
sequent surveys of the premises. 

Twogood v. Hoyt, 42 Mich. 609. 

31. Ancient reputation and possession in regard to 
streets in a town are entitled to more respect in deciding 
on the boundaries of lots than any experimental survey 
that may be afterwards made. 

Ralston v. Miller, 3 Rand. (Va.) 44. 

32. Where lots are sold by numbers and a plat, any 
variance in the distance between known and fixed points 
as found by actual measure on the ground, and the dis¬ 
tance between the same points as laid down on the plat, 
is to be divided between the lots in proportion to the 
respective lengths as laid down on the plat. 

Marsh v. Stephenson, 7 Ohio, N. 3, 264. 

Quinnin v. Reimers, 46 Mich. 605. 

Surplus or shortage in a block is to be divided pro rata 

between the lots. 

Newcomb v. Lewis, 3L Iowa, 488. 

O’Brien v. McGraw, 27 Wis. 446 

33. Where the accuracy of the starting points taken for 
test surveys is merely mattei of speculation, they cannot 
be used to fix a disputed boundary between two lots when 
the dispute arises from a discrepancy which affects all 
the lots in a block, and must therefore be apportioned 
among them. 

Reimers v, Quinnin, 49 Mich. 449. 

34. A resurvey is inadmissible in evidence to show that 
a private boundary is incorrect, if its starting point is 
outside of and does not belong to the immediate plan or 
local system by which the original survey was controlled. 

Burns v. Martin, 45 Mich. 22. 

19* 


270 A MANUAL OF LAND SURVEYING. 

35 . If in running the lines of a grant, one line be found 
which is admitted or proved to be a line of the grant, 
which will run with a variation from the calls of the 
grant, if no other marked lines be found, the other calls 
should be run with the same variation as that found on 
the marked line. 

Sevier v. Wilson, Peck. 146. 

36. Where a deed conveys lots in a town, and refers to 
a plat to identify them, and, in describing their lines, calls 
the points of compass as designated on the plat by its 
lines and angles, a correct survey cannot be based on any 
other system; and although the lines there delineated are 
not conformable to the true meridian, the plat and not 
the compass should govern. 

Bovver v. Earl, IS Mich. 367, 

The remaining decisions under this head, except the 
last four, are of recent issue, and are from the “ Law 
Reporters,” St. Paul, Minn.: 

37. An instruction that, in arriving at a boundary line 
as originally run, natural objects are controlling calls; 
artificial objects, second in importance; course, third, and 
distance, fourth; and that, where there is still uncer¬ 
tainty, that rule should be adopted most consistent with 
the intent of the grant, is correct. 

Luckett v. Scruggs, (Tex.) US. W. 529. 

38. An instruction that the beginning corner of a sur¬ 
vey is of no higher dignity or importance than any other 
corner, and that, “if there are well-known and undisputed 
original corners established upon the ground around the 
survey, they would control the other calls of the survey, 
which are conflicting and contradictory, if there are any 
such,” is correct: 

Luckett v. Scruggs, (Tex.) ll S. W. 529. 

39. Where the beginning corner of a survey is the 
southwest, but the southeast corner is equally well identi¬ 
fied, a charge limiting the jury to finding the unidentified 
northeast corner by the first and second lines from the 



RESURVEYS. 


271 


southwest corner, is erroneous, as the southeast corner is 
of equal importance, unless the line from the former 
corner was actually run and measured, and that from the 
latter not. 

Scott v. Pettigrew, (Tex.) 12 S. \V. 161. 

Lancaster v. Ayres, Id. 163. 

40. An instruction making the importance of an estab¬ 
lished northeast corner, in locating the north and west 
lines of a survey, dependent upon the jury’s belief that 
such western line was not run, is erroneous, as such 
corner has the same weight for the purpose in question, 
whether the western line was run or not. 

Scott v. Pettigrew, (Tex.) 12 S. W. 161. 

41. In the description of lands, as to questions of 
boundaries the rule is settled in Virginia and West Vir¬ 
ginia that natural land-marks, marked lines and reputed 
boundaries will control mere courses and distances, or 
mistaken descriptions in surveys and conveyances. 

Gwynn v. Schwartz, (W. Va.) 9 S. E. 8S0. 

42. The course of the eastern line of the H. tract, as 
given in the original survey made in 1745, was 14 deg. 
east. The course of the western line of the B. tract, lying 
immediately east of the II. tract, as given in the original 
survey made in 1813, was 17 deg. and 15 min. east. The 
western line of the B. tract was made of exactly the same 
length as the eastern line of the H. tract, and the 
beginning point of tbe two lines was the same. The 
difference in the course of the two lines could be 
satisfactorily explained by the change in the position of 
the magnetic needle which had taken place in the time 
intervening between 1745 and 1813. Held, that the two 
lines must be considered as coincident. 

Scott v. Yard, (X. J.) 18 A. 359. 

43. Where neither the corners of plaintiffs’ nor defend¬ 
ants’ land are satisfactorily established, and there is a 
well-established and identified corner of another survey, 
from which, by following course and distance, defend¬ 
ants’ survey can be constructed, such course should be 


A MANUAL OF LAND SURVEYING. 


979 

n 1 Zi 


followed, though the boundaries thus established include 
land within the boundaries of plaintiffs’ junior survey. 

Griffith v. Rife, (Tex.) 12 S. W. 1GS. 

44. A county surveyor, employed to restore the lines 
and corners of adjoining tracts of land according to the 
original government survey, found township corners only, 
then (the other quarter and section corners being missing) 
ran a straight line from one township corner to the other, 
and on this line placed the quarter and section corners, 
but did not take any testimony to ascertain the lines or 
corners of the original survey, did not attempt to prove 
liis lines or corners by re-establishing the missing corners 
from all the nearest known original corners, in all direc¬ 
tions, did not sufficiently regard the field notes, and did 
not, where the original monuments had disappeared, 
regard the boundary lines long recognized and acquiesced 
in. Held , that such a survey is incomplete, and cannot 
be approved as the true and correct determination of the 
boundaries and corners as originally established by the 
government. 

Reinert v. Brunt, (Kan.) 21 P. 807. 

45. Upon an issue as to the location of a line of the 
government survey, evidence of the location of monu¬ 
ments is not overcome by field-notes of the original 
survey, taken at the time of the erection of said monu¬ 
ments or subsequent thereto. 

Ilubbard v. Dusy, (Cal.) 22 T. 214. 

40. As between complicated descriptions of a line divid¬ 
ing two sections or quarter sections, that one is to be 
adopted which is most in conformity with the monument 
established by the government survey. 

Ilubbard v. Dusy, (Cal.) 22 F. 214. 

47. As between different monuments, those best 
identified should prevail, independent of anything in the 
field-notes of the original or any subsequent survey. 

Hubbard v. Dusy, (Cal.) 22 r. 214. 


RESURVEYS. 


273 


48. TV here it is doubtful which of two lines of monu¬ 
ments is the true government line, other things being 
equal, that one is to be so considered which most nearly 
conforms to the field-notes. 

Hubbard v. Busy, (Cal.) 22 P. 214. 

49. On a question as to the true location of a land 
patent, boundaries fixed by reversing the courses and 
distances must govern when found to coincide with the 
natural calls of the patent. 

Ellin wood v. Stancliff, 42 F. 31G. 

50. When the points lixed by reversing the courses and 
distances do not coincide with the natural calls of the 
patent, or the natural calls cannot be identified, then the 
regular courses and distances must govern. 

Ellinwood v. Stancliff, 42 F. 31G. 

51. When a survey calls for the “Dougherty” survey as 
one of its adjoiners, an instruction that if the jury find 
that the “King” is the survey intended by the call for 
“Dougherty,” the former being located, the call would 
furnish “some evidence” of the location of the survey in 
question, is insufficient, as such a finding would locate 
the survey in the absence of marks upon the ground. 

Tyrone Min. & Manuf’g Co. v. Cross, (Pa.) 18 A. 519. 

52. Where no marks are found on the boundaries of a 
survey, and it cannot be located on the ground, evidence 
of the location of junior surveys which call for the lines 
of the elder as adjoiners is admissible, as showing where 
the surveyors upon the ground located such lines. 

Tyrone Min. & Manuf’g Co. v. Cross, (Pa.) 18 A. 519. 

53. Where the court, in an action of ejectment, instructs 
the jury that, “after a survey of blocks had been returned 
and had remained in the land-office 21 years, it was con¬ 
clusively presumed that it was run upon the ground, 
whether marks were found upon the ground or not,” but 
in other portions of his charge repeatedly states the law 
to be that marks made by the surveyor on the ground are 
the first and highest evidence of the true survey, the 
instruction camiot, on the whole, be said to be misleading, 


19 


274 A MANUAL OF LAND SURVEYING. 

as he will be reasonably understood to have charged that 
the presumption in favor of returns of surveys on lile for 
21 years is only applicable to such surveys where no 
monuments or marks on the ground are found to contra¬ 
dict them. 

Grier v. Pennsylvania Coal Co., (Pa.) 18 A. 480. 

54. The exterior of two adjoining interior surveys were 
undisputed. The boundary line between them had never 
been surveyed, but its southern end was marked by an 
oak. North of these surveys were two others. These 
four surveys were originally returned as being of equal 
size, and having one common corner. The northern end 
of the line between these two latter surveys was marked 
by a sugar-maple; which was not directly opposite the 
oak, and it was proved that the northern line of these 
surveys was shorter than the southern line of the others. 
Held , that the boundary line between the two southern 
surveys should run from the oak parallel to the end lines, 
and not diagonally from the oak to the maple. 

Bloom v. Ferguson, (Pa.) 18 A. 488. 

55. Where a dividing line is established between tracts 
of land owned by a county, before purchases are made of 
land on each side of it, and the deeds under which parties 
claim have been made, and are known by the parties to 
have been made with reference to that line, they, and all 
the persons claiming through them, are bound by it. 

Briscoe v. Puckett, (Tex.) 12 S. W. 978. 

56. The northwest corner of a survey was plainly 
marked, and part of the west line was also marked. The 
rest of the survey had apparently not been run on the 
ground, but the southeast corner was ascertainable from 
the field-notes, being located on an established line of 
another survey and at a given distance from an estab¬ 
lished point. The lines of survey as called for in the 
field-notes were correct as to courses, but were too short 
to reach from one of said corners to the other. Held, 
that the survey included all the land between the corners 
bounded by the lines as extended so as to reach from 
one corner to the other. 

Randall v. Gill, (Tex). 14 S. W. 134. 



RESURVEYS 


275 


57. Where a deed describes a lot conveyed as of a cer¬ 
tain width, and a party-wall stands on the south line, the 
north line may be found by measuring the given distance 
north from the middle of such wall. 

Warfel v. Knott, (ra.) IS A. 390. 

58. The statement of the quantity of land supposed to 
be conveyed, and inserted in deeds by way of description, 
must not only yield to natural land-marks and marked 
lines, but also to descriptions in deeds by courses and 
distances. 

Gvvynn v. Schwartz, (W. Va.) 9 S. E. 880. 

59. A call for a lot by the name or number which it 
bears on a plat of the land will prevail over courses and 
distances, and ordinarily over calls for monuments. 

O’Herrin v. Brooks, (Miss.) G So. 844. 

60. Where the descriptions in a deed refer to a survey 
and a map based thereon, making both a part of the deed, 
and there is a discrepancy between the map and the 
survey, the latter will prevail. 

Whiting v. Gardner, (Cal.) 32 P. 71. 

61. The owner of a lot in the city of Rochester, of the 
area of about one-lialf acre, rectangular in form, fronting 
274 feet on a street, and abutting on the rear for the same 
distance on a canal, the location of both, as well as the 
other lines, being undisputed, conveyed a portion, by 
description, of “137 feet front and rear, measuring from 

G. II.’s north line on G. street, and also 137 feet from G. 

H. ’s south line on the canal; being the piece of land 
occupied as a gaixlen by the grantor.” The lot was 
divided by a fence, one side being used as a garden ; the 
fence starting on G. street midway, but striking the back 
line at the canal at a point 19}^ feet from the middle of 
the lot. That fence was not mentioned in the deed. Held , 
that the reference to the garden was too indefinite to 
control the calls for exact distances from known bounds, 
and the divisional point on the canal should be located 
137 feet from G. II.’s line. 

Harris v. Oakley, 7 N. Y. S. 232, 


276 A MANUAL OF LAND SURVEYING. 

62. Plaintiff owned a village lot, No. 124, and a tract of 
land lying adjacent thereto on the south and east sides. 
River street, which lay along a river’s edge, was the 
westerly front of both the lot and the tract. He con¬ 
veyed the tract to defendant, reserving a part thereof, 
beginning at the S. W. corner of the lot; thence south¬ 
easterly, along River street, 32 feet; thence northeasterly, 
“on a line with the southeast corner of lot No. 124,” 10 
rods and 23 links; thence N. to M. street; thence W. 
to the N. E. corner of the lot; thence southwesterly, to 
the S. E. corner; thence to the beginning. Locating the 
beginning point at the S. W. corner of the lot as ap¬ 
peared by the village plat on the easterly side of the 
street, the line passed directly through the S. E. corner 
of lot 124, taking no part of the lot, and thus making the 
reservation wholly within the tract conveyed; but by 
beginning at the river’s edge, on the westerly side of the 
street, on the theory that plaintiff’s property extended to 
the river, subject only to the easement of the street, the 
line would pass through and take part of lot 124. Held 
that the former location of the corner was correct. 

Anderson v. Scott, (Mich.) 42 N. W. 991. 

63. In an action to recover a tract of land lying between 
a slough and a river, plaintiff claimed title by virtue of a 
grant which bounded the land granted by the river, and 
the defendant introduced evidence that the surveyor who 
surveyed the grant meandered the slough instead of the 
river. Held, that, in determining the true boundaries of 
the grant, the sole question was to ascertain exactly where 
the surveyor ran his lines, and, if the jury found that he 
ran the line along the slough, they should find for the 
defendant. 

Allen v, Koepsel, (Tex.) 14 S. W. 151. 

64. Where, in ejectment, a surveyor testified that he 
ran the boundary line in dispute about 1868; that he 
found the original stake of the government survey at the 
section corner, and used it as a starting point; and it 
appeared that about the same time defendant built a 
fence upon this line, which he has ever since maintained 


RESURVEYS. 


277 


—this line must prevail over one surveyed 20 years later, 
when the corner mark was gone, by one who testified 
that he located the section corner by measurements from 
various lines and points, and then by digging found a 
stump which he took to be the original witness, and 
based his survey upon it. 

Carpenter v. Monks, (Mich.) 45 N. W. 477. 

65. The monuments or marks of the surveyor on the 
ground determine the true survey as against calls for 
adjoinders or courses and distances as returned; but, each 
block of surveys being separate and complete of itself, 
the call of a tract in one block for an adjoinderin another 
does not make the monument of the adjoinder the monu¬ 
ment of the later block. 

Grier v. Pennsylvania Coal Co., (Pa.) 18 A. 480. 

66 . Where a boundary line is assented to by the owner 
of a tract of land at a time when there is no dispute con¬ 
cerning such line, and on the supposition that it is the 
true boundary, he is not estopped, on discovering that 
such is not the case, from claiming title to the real 
boundary. 

Schraeder Min. & Manuf’g Co. v. Packer, 9 S. Ct. 3S5. 

67. Continuous and uninterrupted possession, under 
claim of ownership, to the line of a division fence, will 
not bar title, where it appears that such occupation was 
under a belief that the fence was on a true line, and 
without intention of claiming beyond the true line, as 
described in the deeds. 

Skinker v. Haagsma. (Mo.) 12 S. W. 659. 

68 ^ Lands are not surveyed lands by the United States 
until a certified copy of the official plat of survey has been 
filed in the local land office. 

United States v. Curtner, 38 F. I. 

69. One who receives deeds of lots, and conveys to 
others, according to an unacknowledged plat of a town, 
is thereby estopped from denying the sufficiency of the 
dedication for want of the acknowledgment. 

Giften v. City of Olathe, (Kan.) 24 P. 470. 


278 


A MANUAL OF LAND SURVEYING. 


70. Testimony of declarations of a grantor, before the 
execution of a deed, tending to establish a boundary other 
than that made by the deed as construed by the court on 
appeal, is inadmissible, as its effect would be to convey 
land by parole in contravention of the statute of frauds. 

Harris v. Oakley, 7 N. Y. S. 232. 

71. Where a town site w r as surveyed and laid out in 
lots, blocks, streets and alleys, and a plat thereof made 
and lithographed, and distributed among the occupants 
of the town site, and one of the lithographed copies was 
afterwards recorded in the office of the register of deeds, 
but the same was not acknowledged, and the town site 
was pre-empted by the president of the town site com¬ 
pany, and a patent was obtained by him for the benefit of 
the occupants, under the town-site act (5 U. S. St. 657), 
there was a sufficient dedication of the streets and alleys 
of said town, despite the want of acknowledgment of the 
recorded plat. 

Giften v. City of Olathe, (Kan.) 24 F. 470. 

72. A deed conveying land in a town, but “reserving 
streets and alleys according to recorded plat of the town,” 
passes the fee in such streets when such fee was at the 
time held by the grantor subject to the easement of the 
public therein. 

Gould v. Howe, (111.) 23 N. E. G02. 

73. Where surveys of 1837 and 1856 do not agree the, 
former holds. 

Palmer v. Montgomery, 20 N. Y. Rep. 536. 

74. The boundary lines of water lots fronting on a river 
extend into the river at right angles with the thread of 
the stream, without reference to the shape of the shore. 

Clark v. Campau, 10 Mich. 328. 

Bay City Gas Light Co. v. Ind. Works, 28 Mich. 182. 

Twogood v. Hoyt, 42 Mich. 609. 

Norris v. Hill, 1 Mich. 202. 

75. Where a certain distance is called for from a given 
point on a navigable stream to another point on the 
stream to be ascertained by measurement, such measure 
ment must be made by its meanders, and not in a straight 


RESURVEYS. 


279 


line. The same rule prevails when distance is called for 
along a traveled highway. A different rule is sometimes 
adopted when the stream is not navigable. 

When a tract of land is bounded upon a navigable 
stream, the distance upon the stream will be ascertained, 
in the absence of other controlling facts, by measuring in 
a straight line from the opposite boundaries. 

Feople y. Henderson, 40 Cal. 29. 

70. In computing the number of acres in a survey, 
“from,” “to,” and “with” the bank of a stream mean to 
low-water mark. 

Lamb v. Ricketts, 11 Ohio 311. 

1. Alluvium means an addition to riparian land grad¬ 
ually and imperceptibly made through causes either 
natural or artificial by the water to which the land is 
contiguous. It matters not whether the addition be on 
streams which overflow their banks, or on those which do 
not. In each case it is alluvium. 

County of St. Clair v. Livingston, 23 Wall. (U. S.) 4G. 

2. Land formed by alluvium in a river is in general to 
be divided among the several riparian owners entitled to 
it, according to the following rule: Measure the whole ex¬ 
tent of their ancient line on the river, and ascertain how 
many feet each proprietor owned on this line. Divide the 
newly formed river line into an equal number of parts, 
and appropriate to each owner as many of these parts as 
he owned feet on the old line; and then draw lines from 
the points at which the proprietors respectively bounded 
on the old, to the points thus determined as points of 
division on the newly formed shore. 

This rule is to be modified under particular circum¬ 
stances; for instance, if the ancient margin has deep 
indentations or sharp projections, the general available 
line of the river ought to be taken, and not the actual 
length of the margin as thus changed by the indentations 
or projections. 

Deerfield v. Arms, 17 Tick. Mass. 41. 

Jones ct al. v. Johnston, 18 How. (U. S.) 100. 


280 A MANUAL OF LAND SURVEYING. 

3. Under Rev. Stat. U. S. §2396. Held , that in survey- 
ing a lot bordering on a river the water-course becomes 
the boundary, and continues so, no matter how much it 
shifts by accretion, and conveyances of the lot pass all, 
including such accretion to that line. 

East Omaha Land Co. v. Jeffries, 40 F. 38G. 

4. The facts that rapid changes in the banks of the 
Missouri River are constantly going on, and that 40 acres 
have been added to adjoining land, do not overthrow an 
averment of a bill to quiet title to such addition, on the 
ground of accretion, that it was by an imperceptible 
increase, where it was nearly 20 years in forming. 

East Omaha Land Co. v. Jeffries, 40 F. 386. 

5. The rule that owners of land bounded by streams 
are entitled to additions to their land formed by accre¬ 
tion is applicable to the Missouri river, notwithstanding 
the peculiar character of that stream, and of the soil 
through which it flows, whereby changes in its banks are 
great and rapid. 

Jeffries v. East Omaha Land Co., 10 S. Ct. 518. 

6. Where the official plat of the survey of government 
lands shows a river as one boundary of a certain lot, in 
accordance with Rev. St. U. S. § 2395, et seq., a subsequent 
patent for the lot, describing it by number, and referring 
to the plat, on which it is marked as containing a certain 
amount, and deeds, describing the lot by number, pass all 
accretion to the lot up to their respective dates. 

Jeffries v. East Omaha Land Co., 10 S. Ct. 518. 

5. Rules applicable to the United States 
Surveys. — “All the corners marked in the surveys 
returned by the surveyor-general shall he established as 
the proper corners of the sections or subdivisions of 
sections which they were intended to designate.” 

“ The boundary lines actually run and marked in the 
surveys returned by the surveyor-general shall he estab¬ 
lished as the proper houndary lines of the sections or 
subdivisions for which they were intended; and the length 
of such lines as returned shall he held and considered, as 
the true length thereof.” 


REStTRVEYS. 


281 


The preceding quotation from section 2396 of the Re¬ 
vised Statutes of the United States, settles all questions 
in regard to any change in the corners, lines or measures 
of the government survey. They are thereby made un¬ 
changeable, the statute thus emphasizing the common 
law, which holds the same doctrine to be true of all orig¬ 
inal surveys after the land has been conveyed in accord¬ 
ance with them. Hence, in making resurveys, the surveyor 
must find, if possible, the original corners, and make his 
courses and distances agree with those of the United 
States survey. 

The following points have been decided by the courts 
with reference to these surveys : 

Rule 1 . — The original surveys by which the govern¬ 
ment sold its land and conveyed it to the purchaser 
establish the rights of the parties as to the boundaries 
ISTo line which will vary the rights thus acquired can 
afterwards be established without the consent of all 
parties. 

May v. Baskins, 12 S. and M. (Miss.) 42S. 

2. Land sold under the United States surveys pass ac¬ 
cording to the description of the legal subdivisions, 
whether those subdivisions contain the legal quantity or 
not, more or less. 

Fulton v. Doe, G Miss. 751. 

3. Each section or a subdivision of a section is inde¬ 
pendent of any other section in the township and must 
be governed by its marked and established boundaries. 
Should they be obliterated, a last recource must be had to 
the best evidence that can be obtained showing their 
former situation and place. 

Lewen v. Smith, 7 Port (Ala.) 428. 

4. Field notes must yield to actual monuments erected 
by the original surveyor. They are only to be relied on 
as evidence to assist in finding the exact situation of the 
monuments. 

McOlintock v. Rogers, 11 Ill. 279. 



282 


A MANUAL OF LAND SURVEYING. 


5 Monuments found at the two extremes of a township 
line are entitled to no more controlling influence in de¬ 
termining the actual location of an intermediate line 
than the section corners established along the line. All 
original monuments established in connection with the 
field notes and plats must be referred to in order to define 
the locality of the line. 

McClintock v. Rogers* 11 Ill. 279. 

6. The corners established by the original surveyors of 
public lands by authority of the United States are con¬ 
clusive as to the boundaries of sections and divisions 
thereof; and no error in placing them can be corrected by 
any survey made by individuals or a state surveyor. 

Arnier r. Wallace, 28 Miss. 55G. 

In ascertaining the lost corner of a section, recourse 
must be had to the unobliterated marks of the original 
survey, the field notes and plats and subsequent surveys 
made under their guidance. If only a portion of one of 
the boundary lines leading to the lost corner on a town¬ 
ship line has been obliterated, the remaining portion must 
be considered established as marked, and the corner must 
be presumed, in the absence of evidence to the contrary, 
to be at the point where the marked line if continued 
would intersect the township line But if the lost corner 
is proved to have been at another point, the lost portion 
of the boundary must be ascertained by running a straight 
line from the point where the marks disappear to that 
corner. 

Billingley v. Bates, 30 Ala. 378. 

7. In determining the line between the quarters of a 
section, the quarter post established by the government 
surveyors must govern in all cases where its location can 
be ascertained 

Vroman v. Dewey, 23 Wis. 530. 

Britton v. Ferry, 14 Mich. 53. 

8. In re-establishing a lost quarter post on a section 
line, any difference in the length of such line by actual 


RESURYEYS. 


283 


measure as compared with that indicated by the govern¬ 
ment survey should be divided between the parts in pro¬ 
portion to their respective lengths as shown by that 
survey. 

Jones v. Kimble, 10 Wis. 429. 

9. If the distance between recognized government cor¬ 
ners as originally established overruns or underruns that 
given in the field notes, it should be divided pro rata be¬ 
tween the intervening sections. The original field notes 
should be the main guide. Section lines being frequently 
deflected, the true corners must be tested by east and west 
distances from the recognized government corners yet 
standing in the same township as well as by north and 
south distances. 

Martz v. Williams, 67 Ill. 306. 

10. Unknown corners must be found by the corrobora¬ 
tive testimony of all known corners with as little depar¬ 
ture as may be from the system adopted on the original 
survey, without giving preponderance to the testimony 
of anv one monument above another. 

In re-establishing lost corners between remote corners 
of the same survey, when the whole length of the line is 
found to vary from the length called for; we are not per¬ 
mitted to presume that the variance arose from the de¬ 
fective survey of any part, but must conclude in the 
absence of circumstances showing the contrary that it 
arose from the imperfect measurement of the whole line, 
and distribute such variance between the several subdi¬ 
visions of the whole line in proportion to their respective 
lengths. 

Moreland v. Page, 2 Clarkes, Iowa, 139. 

11. Quarter posts of the government survey are to be 
as much respected as the corners of townships or sections 
however distant from the center line. 

Campbell v. Clark, 8 Mo. 558. 

12. There was a mistake in the government survey of a 
section by which the quarter section line and the meander 


284 A MANUAL OF LAND SURVEYING. 

line of a river were shown on the official plat to be one 
and the same line, being the boundary line of the frac¬ 
tional lots. As a matter of fact they were a considerable 
distance apart. There was no question as to the location 
of the quarter section corners. In a suit to determine 
the ownership of the land between the quarter section 
line and the river, it was held that the quarter section 
line should be adhered to as the more certain call, and 
that where the lines of a survey can be run from well 
ascertained and established monuments, they are to con¬ 
trol and govern a description delineated on a plat, al¬ 
though the quantity in the fraction fell short of the 
amount laid down in the plat about as much as there 
was land contained between the quarter line and the 
river. 

Martin v. Carlin, 19 Wis. 454. 

13. When a deed designates the land conveyed as one 
of the subdivisions known in the United States survey, 
as, for instance, a quarter, half-quarter or quarter-quarter 
section, the presumption is that the parties intend that 
the tract shall be ascertained in the same manner as is 
done in the government surveys. 

Not so, where the deed conveys a tract of land not 
known in that system of surveys, as, for instance, the 
east half of a lot, or of a quarter-quarter section. 

Cogan v. Cook, 22 Minn. 142. 

14. The defendant sold the north half of a lot which is 
bounded on the west side by the Au Gres river. But the 
river is not straight at this point, and the north line of 
the lot is longer than the south line. 

The bill demands the north half ol the lot, and the 
north half must mean the north half in quantity divided 
from the remainder by an east and west line. 

An Gres Boom Co. v. Whitney, 2G Mich. 44. 

15. It is a question of fact to be determined by all the 
surrounding circumstances whether the land between the 


RESURVEYS. 285 

meander line and the shore of the lake or water course is 
included in the survey. 

Shoemaker v. Hatch, 13 Nev. 267. 

16. The lines run to divide sections into halves and 
quarters, if erroneous, may be corrected, for they are 
subdivided by law; and if the officer in running 1 the sub¬ 
division line makes a mistake, it can be corrected by run¬ 
ning the line according to law. 

Nolin v. rainier, 21 Ala. 66. 

17. An original township was divided into sections “ by 
running through the same, each way, parallel lines at the 
end of every two miles, and making a corner at the end 
of every mile,” and afterward a supplemental survey was 
made under a subsequent statute, which directed that 
these two mile blocks should be subdivided by running 
straight lines from the corners thus marked to the oppo¬ 
site corresponding corners. Held, that where the original 
mile corners in a certain block can be clearly identified, 
the courses of lines of subdivision within the block can¬ 
not be determined by proof of monuments, blazes, or 
other witness marks found in other blocks in the town¬ 
ship. 

Ginn v. Brandon, 29 Ohio St. 656. 

18. When a navigable stream intervenes in running the 
lines of a section, the surveyor stops at that point, and 
does not continue across the river. The fraction thus 
made is complete, and its contents can be ascertained. 

Therefore, when there is a discrepancy between the cor¬ 
ners of the section as established by the United States, 
and the lines as run and marked, the latter do not yield 
to the former. 

Lewen v. Smith, 7 Port. (Ala.) 428. 

19. In government surveys, the line actually run by the 
government surveyors is the true line. 

Goodman v. Myrick, 5 Oregon, 65. 

20. In a case where the township lines had been run 
and marked by the United States survey, but the field 


286 A MANUAL OF LAND SURVEYING. 

notes of the subdivision lines were fraudulent and re¬ 
jected by the surveyor-general, because incorrect, no 
proper survey of them having been made, it was held 
that the line between sections one and two must be ascer¬ 
tained by running a straight line from the corner of the 
sections established on the exterior line of the township 
to the corresponding corner on the opposite side of the 
township. 

Hamil v. Carr, 21 Ohio St. 258. 

21. Where the initial point in the description of prem¬ 
ises in a deed is the southeast corner of the north half of 
the southeast quarter, fractional, of a section, and the 
quarter-section is made fractional by a meandered lake 
so situated as to cover the eastern and central portions 
thereof; and the parcel described was carved out of the 
north half within a year after the same was patented, 
the southeast corner in question is construed to be the 
point which constituted the southeast corner of the land 
as it was surveyed out and platted by the government, 
which located it on the meandered line of the lake. The 
fact that the waters of the lake have since receded can¬ 
not change the boundaries as previously located. 

Verplanck v. Hall, 27 Midi. 79. 

22. Extending fractional lots beyond quarter lines: 
Etheridge and Stone were the original settlers, pre-empt- 
ors, and purchasers of fractional section 22. Etheridge’s 
patent called for “ the S. W. }£ of Sec. 22, containing 92.67 
acres.” Stone’s patent called for “S. E. subdiv. Qr. Sec. 
22, containing 110.50 acres.” These two descriptions were 
in controversy in 

Brown’s lessees v. Clements, 3d How. G50. 

In the figure (page 287) the full lines show the frac¬ 
tional section as it was returned on the ofiicial plat. The 
dotted lines show the quarter lines as they would have 
been if the section had been full. 


RESUKVEYS. 


287 


On the part of the grantees of Etheridge two claims 



were set up. One was that 
under the pre-emption laws 
Etheridge was entitled to a 
full quarter section of land. 
The other was that, as his 
deed called for the S. W. 


and the fractional section 
was of such size and shape 
that a regular southwest 
quarter could be laid out 
from it, he was entitled to 
it, and that the action of the 


Fig. 71 


Surveyor General in returning irregular subdivisions of 
the section, when he could have made one regular quarter 
section out of it, was contrary to law, and therefore void. 
The Supreme Court by a bare majority upheld these 
claims and decided the case on those grounds. 

The case of Brown’s lessees v. Clements was decided in 
1845, several of the judges strongly dissenting from the 
decision. In 1858 the same tract of land came in question 
again. 

Gazzam v. Phillips’ lessee and others, 20th Howard 372. 

Speaking of the sales to Stone and Etheridge, the Court 
says: 

“ The sales in each case were made in conformity with 
the plat of the survey then on file in his oftice,” etc. 

“ We deny altogether the right of the court in this ac¬ 
tion to go beyond these terms thus explicit and specific 
and under a supposed equity in favor of Etheridge, 
arising out of the pre-emption laws, to the whole of the 
southwest quarter—enlarge the description in the grant, 
or more accurately speaking, determine the tract and 
quantity of the land granted by this supposed equity 
instead of by the description of the patent. 

“We are not satisfied that there was any want of power 
in the surveyor general in making subdivisions of this 








288 A MANUAL OF LAND SURVEYING. 

section according to the plat and in conformity with 
which the sales of the lands in dispute were made. 

“The Act of 1820 provides that fractional sections 
containing 160 acres and upwards shall in like manner, 
as nearly as practicable, be subdivided into half quarter 
sections under such rules and regulations as may be 
prescribed by the secretary of the treasury. 

“The secretary of the treasury, on the lOtli of June 
following the passage of the act, issued regulations 
through the commissioner of the land ottice, directing 
fractional sections containing more than 160 acres to be 
divided by north and south or east and west lines, so as 
to preserve the most compact and convenient form. This 
section was divided by a north and south line according 
to these instructions. The question came before the 
secretary of the treasury and before us in 1837, and the 
construction first given and the practice of the surveyor 
general under it confirmed. Attorney General Butler in 
a well considered opinion observed: ‘If congress had 
intended that fractional sections should at all events be 
divided into half quarter sections when their shape per¬ 
mitted the formation of such a subdivision, I think they 
would have said so in explicit terms, and that the discre¬ 
tionary power entrusted to the secretary would have been 
plainly confined to the residuary parts of the section. 
And further that the clause in the first section of the act 
of 1820, concerning fractional sections containing less 
than 160 acres (which are not to be divided at all) is 
decisive to show that congress * * did not deem it 

indispensable that regular half quarter sections should in 
all practicable cases be formed by the surveyors. On 
the contrary, it shows that they preferred a single tract 
though containing more than 80 acres to small incon¬ 
venient fractions.’” 

The court adds: “We entirely concur in this construc¬ 
tion of the act,” and further goes on to say: “The only 
difficulty we have had in this case arises from the cir¬ 
cumstance that a different opinion was expressed Dy a 


RESURVEYS. 289 

majority of this court in the case of Brown’s lessees v. 
Clement, 3 How. 050. 

“ It is possible some rights may be disturbed by refusing 
to follow the opinion expressed in that case, but we are 
satisfied that far less inconvenience will result from this 
dissent than by adhering to a principle which we think 
unsound and which in its practical operation will unsettle 
the surveys and subdivisions of fractional sections of the 
public land running through a period of some 38 years. 
We cannot adopt that decision or apply its principles in 
rendering the judgment in this case.” 

10. Quarter posts on section lines where there are double 
sets of section corners: “ Quarter section corners are not 
required to be established on the west boundary of the 
western tier of sections in a township, nor on the north 
boundary of the north tier of sections in a township south 
of and bordering on a standard parallel. The resurvey 
of township , standard, or base lines, by the deputy sur¬ 
veyor for the purpose of establishing such quarter-posts, 
is unnecessary and will not be paid for.” 

Instructions to surveyors-general by Commissioner Edmunds, p. 0. 

11. “ Range lines are run north or south from the base 
line, and corners for sections and quarter sections are 
established thereon at every mile and half mile for the 
sections and quarter sections on the west side of the line, 
but not for those on the east side” On township lines 
“the corners of sections-and quarter sections are estab¬ 
lished at every 80 and 40 chains for the sections and 
quarter sections on the north side of the line, but not for 
those on the south side” 

Instructions to Deputy Surveyors of the United States for the 
district of Illinois and Missouri, 1856, p. 50. 

6. Decisions of the General Land Office 
with reference to Mineral Surveys.— Plats and 
field notes: Of surveys of mining claims, required to 
disclose all conllicts with prior surveys, giving areas of 
all conllicts. 


290 A MANUAL OF LAND SURVEYING. 

In future, surveyor-general will use no coloring on 
plats. 

Com’r. (N.) Nov. 1G, 1882. Circular. 

Location {of mine): Must be marked on the ground so 
that its boundaries can be readily traced. 

N. Noonday M’g Co. v. Orient M’g Co., G Saw.,C. C., 299; Myers ct nl. 
v. Spooner et al ., 55 Cal. Ii. 257; Gleason v. N. White M’g Co., 18 Nev. 
R., 443; Southern Cross G. and S. M’g Co. v. Europa M’g Co., 15 id., 383. 

Surface line: Agreement by adjoining claimants, fixing 
surface boundary line between them, must be construed 
as extending such line downward, through the dips of 
the vein or lode, to the earth’s centre. 

Richmond M’g Co. v. Eureka M’g Co., 103 S. C., 389. 

Bearings and distances must be given in a survey, from 
the respective survey corners to the location corners, and 
the same must be shown on the plat. 

Survey: Of a mining claim should show location of 
all improvements of a municipal nature, as blocks, alleys, 
etc. 

Scc’y Dec. 18, 1880, and Feb. 3, 1881. Little Nettie Lode. 

7. Descriptions in Deeds. — Surveyors are fre¬ 
quently required to make surveys for the purpose of fur¬ 
nishing a description of the land to be conveyed. Every 
surveyor of experience is familiar with the many diffi¬ 
culties encountered in correctly locating boundary lines, 
caused by defective, false or impossible descriptions in 
the deeds. The description is the controlling guide to the 
surveyor in locating a man’s possessions on the ground, 
hence it is important that it should be clear, distinct and 
harmonious in its terms. 

Where land is conveyed in the regular subdivisions of 
the United States survey, little difficulty will be met in 
writing a correct description. The main caution to be 
observed is to avoid the common clerical error of using 
the wrong letter or word, such as north instead of south, 
or east instead of west, thereby locating the deed in a 
different place from which it was intended. Scrutinize 


RESUIIVEYS. 


291 


the description closely to see that no such error is made, 
and write plainly, so that no one need make a mistake 
in reading or copying the description. A great many of 
these mistakes are caused by bad penmanship. 

Similar remarks apply to the description of land by 
plat, where only clerical errors are likely to be made. 

It is in the description “ by metes and bounds ” and by 
courses and distances, that greatest care should be taken. 

Do not use two descriptions if one will clearly describe 
the land. Avoid surplusage and conflicting descriptions. 
If after writing a description it is found necessary to 
explain it, lay it aside and if possible write a description 
that does not need explanation. 

Let the starting point be well defined and permanent, 
so that there need be no difficulty in locating it at any 
time in the future. A striking example of a disregard of 
this principle was brought to the attention of the writer 
when he was called to locate the boundary lines of several 
lots in a village. The descriptions all referred back to 
a small cherry tree as a starting point. The lines had 
never been marked on the ground even by fences, and the 
cherry tree had been gone so long that no one could be 
found who could remember that there ever was such a 
tree. 

N ot only the starting point but as many of the angles 
in the boundary as possible should be described by some¬ 
thing permanent and definite on the ground. This is 
of prime importance. Let it be the plainest and most 
permanent that the nature of the case permits. 

If the courses are given by compass bearings, state 
whether they refer to the magnetic or some other merid¬ 
ian. This is put in the form of a statement of the decli¬ 
nation of the needle, written for example, Var. 4° 20 7 E. 
By this it is understood that the magnetic meridian 
makes an angle of 4° 2CK to the east of the meridian of 
the survey, it was formerly a custom to refer all lines to 
the magnetic meridian. Since the adoption of the system 
Of the United States Land Surveys it has become a 






292 


A MANUAL OF LAND SURVEYING. 


custom, especially in that part of the country surveyed 
under that system, to refer all surveys to the true merid¬ 
ian, or what was supposed to be so. As time has passed 
and old descriptions have been retained in the deeds 
conveying the land from owner to owner, it has become 
impossible in thousands of cases to tell what meridian 
controls the description. Hence we see the prime 
importance of permanent monuments describing the 
boundaries, and of describing the meridian of the survey. 
If we must needs figure out courses from the change in 
direction of the needle, let us have something definite to 
start from. 

Do not describe a boundary solely by reference to the 
boundary of the adjoining tract, if it can be avoided 
without error. Such a description requires the finding of 
the description of the adjoining tract whenever a survey 
is made, and may cause great delay and trouble before 
the correct definite description can be found. The writer 
knows of a case where the only description of the bound¬ 
ary line between two village lots in either deed is by a 
reference to the other: A.’s land is bounded on the east 
by B.’s land, and B.’s land is bounded on the west by A.’s 
land—nothing more. 

If a boundary line is not intended to be a straight line, 
but to follow a fence, a wall, a hedge or a stream, say 
so in the description. Make everything clear, definite, 
concise and consistent throughout, so that a surveyor 
having the description in the deed can locate the boun¬ 
daries on the ground, without having to hunt up descrip¬ 
tions from other deeds. 

8. Illustrations.—1. “ The east half of the northeast 
quarter of Section 16, Township 2 south , Range 10 west.” 

The United States land department in selling land in 
regular subdivisions of non-fractional sections does not 
state the quantity in the patent. It is quite customary 
in later conveyances to add something like the following: 
“containing 80 acres, more or less, according to the 
United States survey.” Nothing is gained by the addi- 


RESURVEYS. 


293 


tion. There is a good deal of useless verbiage and repe¬ 
tition in deeds, the only effect of which is to add to the 
expense of making out and recording them. 

2. “ The north fractional half of the northeast frac¬ 
tional quarter of Section 3 , Township 3 south, Range 9 
west , containing 98.72 acres, according to the official plat 
of the United Stales Survey .” 

The area of fractional lots is stated in the United 
States patents. The word fractional is used and the 
area given to show that the land is conveyed according 
to the system of the United States survey. Without 
them the description would convey the aliquot part of 
the entire area of the section in the same manner as 
Description No. 1. 

3. “ The south fraction of the southeast quarter of 
Section 28, Township C, north, Range 3 west, containing 
117.85 acres.” 

Sections are made fractional by streams, lakes and 
reservations, making fractional lots of all manner of 
sizes and shapes. The land department attaches small 
outlying fractions to the adjacent larger ones, and sells 
the whole under one description, which takes its name 
from the larger lot. The above description might con¬ 
tain land attached from the southwest quarter. Such 
descriptions do sometimes contain land attached from 
other sections, and even from other townships. The 
official plat of the section show r s precisely what land is 
included in the description. 

4. “A piece of land twenty feet wide off from the east 
side of Lot 99 of the lithographed plat of the village of 
Kalamazoo.” 

A description like the above sometimes leads to contro¬ 
versy. Suppose the original survey by which the lots 
were laid out, was made with a long chain, as it was in 
Kalamazoo, and that there was a surplus in the lot. The 
purchaser might claim that he was entitled under the 
common law to his proportional share of the surplus, 
while the seller, if he owned the balance of the lot, 
might claim it all as his own. Such questions do fre- 






291 * A MANUAL OF LAND SURVEYING. 

quently arise, and it is better to settle them at the outset, 
by putting it definitely in the description what is meant. 
In the above case suppose the recorded width of the lot 
to be sixty feet; then a description calling for the “ east 
one-third of Lot 99 ” would show clearly that any surplus 
or shortage in the lot was to be divided, while a descrip¬ 
tion reading “20 feet off the east side of Lot 99, etc., as 
surveyed by F. II., May 22nd, 1883,” would show that the * 
later surveyor’s measure was to govern. The care and 
accuracy of measurement of land in cities keeps pace 
with its increase in value, and as a careful, accurate 
measure cannot be expected to agree with a careless, 
inaccurate one, it is best to settle such questions in 
advance, as far as possible. 

5. “Commencing at a stone with a hole drilled in it, set 
in the east and west quarter line of Section IS, Township 
4 south, Range 40 west, 22 elm,ins crust, of the range line, 
from which stone a 

White oak 16 inches diameter, hears 8.28° W., 62 links 
distant , and running thence (Far. 2° 40' E., at 40 A. M., 
June 42th, 4SS0), north 22° east 42.00 chains to a stone 
marked with a cross, set in an angle of a hedge y 

Thence east along the hedge 8.00 chains to an iron stake 
of 4% inch gas pipe, driven on west hank of a ditch; 

Thence south along the hank of the ditch 5.00 chains to 
an iron stake of gas pipe driven in the hank where the 
ditch turns east; 

Thence south 22° west 6.61 chains to a stake set in the 
quarter line, from which a 

Burr Oak 42 in. di. hears N. 46° E., 26 Iks. distant, 

Burr Oak 48 in. di. hears 8. 46° E., 51 Iks. distant; 

Thence west along the quarter line 40.21 chains to the 
place of beginning .” 

This is given as a sample of a description by metes and 
bounds such as a surveyor may furnish under the ordinary 
circumstances when called on to make a survey for that 
purpose, and such as he or any other surveyor would have 
no trouble in locating on the ground at any future time 1 
so long as any of the monuments or bearing trees could 
be found. 


RE-LOCATION OF LOST CORNERS. 


295 


CHAPTER XT. 

RE-LOCATION OF LOST CORNERS. 

The general principles to be observed in re-locat¬ 
ing lost corners are laid down in the Supreme Court deci¬ 
sions which have already been quoted. 

A corner is not lost so long as its position can be deter¬ 
mined by evidence of any kind without resorting to sur¬ 
veys from distant corners of the same, or other surveys. 
Often after making a survey from a distant corner, the 
surveyor will come upon some traces or evidence which 
will enable him to determine the true position of the 
corner he is seeking. It is an uncertain way at the best 
to locate corners by running lines and measuring from 
distant corners, and should only be resorted to in absence 
of better proof of the original location of the corner 
sought. 

It will sometimes happen that the exact spot where a 
lost corner stood cannot be found or shown by evidence, 
but it can be proved that it stood within certain limits. 
In these cases, which are not rare, there is no question 
but that the corner should be placed at that point within 
the known limits which best agrees with all the evidence 
in the case. 

Failing of better evidence by which to determine the 
location of a lost corner, we may next resort to the fol¬ 
lowing methods: 

General Rule. —Retrace the known lines of the de¬ 
scription and find how the lengths and directions of these 
lines by your survey agree with those of the same lines 
as laid down in the original description. Then run the 


296 


A MANUAL OF LAND SURVEYING^ 


unknown lines and place the lost corners so that they 
will bear the same relation to the known lines and cor¬ 
ners as they are required to do by the description of the 
original survey. 

Example. —The four lines of a description are as fol¬ 
lows: 

1. North 7° east 12.00 chains. 

2. South 83 n east 0.00 “ 

3. South 7° west 12.00 “ 

4. North 83° west 6.00 “ 

The first line and its termini are known. We retrace 
that line and find by our survey that it runs north 7° 30' 
east and 12.24 chains. 

We would then run the remaining lines, making them 
as follows: 

2. South 82° 30' east 6.12 chains. 

3. South 7° 30' west 12.24 “ 

4. North 82" 30' west 6.12 “ 

Or the compass may be set on the known line and the 
vernier so adjusted that the reading of the needle shall 
be the same as that given in the original description and 
the remaining lines run accordingly. 

2. Re-location of Lost Corners of the United 
States Survey. 

Rule 1.—On baste lines, correction parallels, township 
and range lines. Restore the lost corner in line between 
the nearest known corners on the same line and at dis¬ 
tances from them proportional to those laid down in the 
field notes of the government survey. 

This rule supposes the original line to have been a 
straight line. As a matter of fact this is frequently not 
the case. If there is reason to suspect the line to have 
angles in its course, measures from known corners to the 
right and left of the line will aid in determining its true 
position. 

Rule 2. —Lost closing- section oorners upon a town¬ 
ship or range lino, where the closing distance from the 


RE-LOCATION OF LOST CORNERS. 


297 


adjacent corners is not given in the field notes should be 
restored by prolonging the known portion of the line to 
its intersection with the township or range line. 

Rule 3. Lost interior section corners should be 
restored at distances from the nearest known corners, 
north, south, east and west, proportional to those laid 
down in the field notes of the original survey. 

This rule supposes that the measurements of the origi¬ 
nal survey were uniform on the several adjacent sections. 
This is frequently not the case, and it will be well for the 
surveyor to compare his chaining on each section with 
the original measure between known corners of the same 
sections, choosing by preference those lines which on the 
government survey were measured next previous to the 
portion of the line closing on the lost corner. 

Rule 4. — Lost township corners, when common to 
four townships, are to be restored in a similar manner to 
interior section corners, Rule 3. When common to only 
two townships, they are to be restored according to Rule 1. 

Rule 5.— Lost quarter section corners are to be re¬ 
stored in line between the section corners which stand on 
the same line and at distances between them proportional 
to those returned in the field notes of the government 
survey. 

Rule G. — Lost meander corners are to be restored by 
running the line from the nearest known corner the di¬ 
rection and distance called for by the notes of the orig¬ 
inal survey. When a portion of the line leading to the 
meander corner is known, it should be prolonged in the 
same direction. When no portion of the line is known’ 
the surveyor will have to use his own judgment as to 
what method under the circumstances of the case will 
most nearly retrace the original line to the corner. 

There is no rule which will rigidly and inflexibly apply 
to all cases for restoring lost corners and boundary lines 
except this—that the aim of the surveyor should always 






298 


A MANUAL OF LAND SURVEYING. 


be to find the exact spot where the original corner or line 
was located. The thing to find out is not where the cor¬ 
ner or line ought to have been , but where it actually was. 

There are many cases in which other methods for re¬ 
storing any of the corners mentioned will prove more 
satisfactory than the rules heretofore given. 

For instance, a half-quarter post properly planted at a 
time when both the section and quarter-section corners 
adjacent were known, may be used in restoring either of 
these corners when lost, by prolonging the line over the 
known corners and doubling the distance. Any other 
intermediate corner whose location is definitely known 
may be used in a similar manner. On a similar principle, 
the Supreme Court of Illinois decided in the case of Noble 
v, Chrisman (88 Ill. 18G) that the northwest corner of sec¬ 
tion 19 could, in that instance, be better determined by 
tracing the section lines from known corners east and 
west of the range line to their intersection with that 
line, and measuring the jog between the corners, than it 
could by prorating six miles of the range line. 

Most of the difficulties which the surveyor has to con¬ 
tend with in restoring lost corners arise from errors made 
in the original survey, or in the field notes thereof. He 
should bear in mind that errors in the original survey 
cannot be corrected by him. In any case of a lost corner, 
find as many of the adjacent corners of the original sur¬ 
vey as possible, according to the best evidence that can 
be had to prove their exact location. Having done this, 
the others may bo found according to the rules already 
laid down. But do not give up a corner as lost while any 
means of finding its exact location are left untried. There 
is great virtue in a pick and shovel intelligently applied 
to the finding of corner posts and monuments. This is 
very important, as it is very difficult, if not impossible, in 
many cases, to re-locate a lost corner in the exact position 
it originally occupied, by surveys from distant corners. 
The following extracts from a paper read by the author 


RE-LOCATION OF LOST CORNERS. 


299 


before the Michigan Association of Surveyors and Engi¬ 
neers, treat more fully of the application of the foregoing 
principles to finding corners of the United States survey 
in those regions where wooden posts were planted for 
corner monuments: 

“ It often happens that one surveyor will fail utterly in finding the 
marks of an origina. corner, while another, more apt in discovering 
the evidences, will strike upon it readily. These evidences are of vari¬ 
ous kinds, some of which it is the principal aim of this paper to dis¬ 
cuss. 

I take it that the best possible evidence of the location of an orig¬ 
inal corner is the monument fixed at that corner when the survey was 
made. (Vide McClintock v. Rogers, 11 Ill. 279; also Gratz v. Hoover, 16 
Penn. State Rep. 232; 16 Ga. 141.) After this come witness trees, fences, 
distant corners of the same survey, and the testimony of persons. 

All these latter kinds of evidence only go to corroborate the first, 
and may take the place of the first only so far as they may any of them 
seem to have weight in any particular case. 

Many of the corners of the United States survey were marked by 
planting a post or stake in the ground. These stakes had notches cut 
in them, were squared at the top, and set in certain regular positions 
fn the ground. These marks tended to distinguish them from other 
stakes that might chance to be driven in the ground for any purpose. 
When trees stood conveniently near, two of them were marked, and 
their directions and distances from the corner were given in the field 
notes. When no trees were near, a mound was sometimes raised about 
the post. 

Some of the posts have been entirely destroyed, but the bottoms of 
a great many of them still remain, much decayed, but plainly visible 
when the surface earth is removed from about them. 

To find them, careful manipulation is required. The surveyor first 
determines as nearly as he can, from extrinsic evidence, the point 
where the corner post should be looked for. He then, with a shovel, 
spade or hoe, carefully removes the surface earth, a little at a time, 
being particular not to strike deep at first into the earth at the level as 
it was when the stake was set. The best and sometimes the sole evi¬ 
dence of a corner has often been destroyed by an ignorant person 
striking deep into the ground, expecting to find a sound stake, and 
casting away the decayed wood and filling up the hole of a rotten one 
without observing it. If the surveyor is looking in the right place, and 
the earth has not been previously removed, he will soon come upon the 
object of his search; but he must be careful lest he mistake it. If the 
soil is a stiff clay, packed hard, as in a road, or covered with a sward, 
ho will presently find a hole of the size and shape of the stake which 


300 


A MANUAL OF LAND SURVEYING. 


made it. This hole Avill contain the decayed wood of the stake, and a 
marking pin may he readily thrust to the bottom. By carefully scrap¬ 
ing or cutting away the earth from the top, or cutting down at one side 
of the hole, its size, shape and direction may he readily discovered. 
Thus it often happens that the position of a corner is as well and sat¬ 
isfactorily marked by the decayed stake as it was by the sound one. It 
sometimes happens that new stakes have been driven beside the orig¬ 
inal stake, so that several different ones will be found by the surveyor. 
He will seldom have any difficulty in deciding which is the true corner 
by its appearance, for the first stake will he more completely decayed 
and of a darker color. 

As a rule, it will be driven deeper and straigliter down than the 
newer stakes. Then, too, the original stakes were generally round, 
being cut from whole timber, Avhile the later ones were often cut from 
rails or other split timber, the sharp corners of which can be readily 
seen in the holes made by them. 

There is thus in the appearance of the stakes of the United States 
survey such peculiarities and such likeness to each other, even when 
far gone in decay, that the experienced surveyor will be impressed 
with the appearance of truthfulness pervading them, and will seldom 
be deceived. This appearance of truthfulness about a stake, which 
to a surveyor is one of the most valuable parts of the testimony of 
these silent witnesses, is something that courts and juries can seldom 
take cognizance of, because, first, they speak in a language that courts 
and juries do not understand, and secondly, the evidence is itself de¬ 
stroyed by the surveyor in the taking, and does not come before court 
or jury in all its freshness, truth and purity. These decayed stakes 
may be best observed in the light-colored subsoil after the black sur¬ 
face mould has been removed. In sandy soil, the cavity made by the 
stake is gradually filled by the falling sand as the wood decays, but 
rotten wood discolors the sand so that where it has not been disturbed 
the position, size and shape of the stake may be readily traced. In 
the black muck of our marshes and river bottoms it is more difficult 
to distinguish the stake near the surface, but as the ground is soft and 
wet the stakes were driven deep, and we may sometimes find in the 
wet, peaty subsoil the bottom of the stake so perfectly preserved that 
even the scratches made in the wood by nicks in the axe are plainly to 
be seen. When the stakes are constantly wet, they do not decay. 

Next we consider the bearing or witness trees. These are marked 
and their directions and distances noted, in order to assist in finding 
the corner posts set on the survey. These bearing trees are marked 
with a blaze and a notch near the ground on the side facing the corner. 
The measures were taken from this notch. At this time most of the 
living witness trees have grown to such an extent that only a scar re¬ 
mains in sight, to indicate the point where the notch was cut. In order 


RE-LOCATION OF LOST CORNERS. 


301 


to get at the notch, the superincumbent wood, which is in some cases 
a foot in thickness, will have to be cut away. It will not often be 
necessary to do this, as we can come sufficiently near the correct point 
to find the stake without it. But if the stake has been destroyed, or 
there are several stakes near, we shall need to be exact, and measure 
from the notch. If the tree has been cut down, and a sound stump 
remains, the marks will be easily exposed. Sometimes the mark is 
gone, but a part of the stump is left. At others the stump is gone, but 
a dish-like cavity remains in the earth to show where the tree once 
stood. We can almost always find under and around these cavities 
places where the large roots have penetrated the subsoil, and thus be 
able to locate within a foot or so the position of the bole of the tree 
when standing. In looking for a corner post, we may frequently as¬ 
sume for the time being that a certain stump or a cavity where a tree 
had stood was the stump of or the place occupied by a bearing tree. 
If we then measure the required direction and distance, and find a 
stake, we may reasonably conclude that our assumption was correct. 
Such assumptions are frequently of great assistance in finding corners. 
There may be, and I know there are cases, where the original corner 
stakes have been destroyed, and can be more nearly restored to their 
original position by measurements from old stump bottoms or holes in 
the ground than in any other way. But bearing trees, however good 
their condition, are by no means infalible witnesses as to the location 
of a corner. Mistakes in laying down their direction or distance, or 
both, are not rare. (See McClmtock v. Rogers, 11 Ills, 279.) A direc¬ 
tion may be given as north instead of south, east instead of west, or 
vice versa. The limb may have been wrongly read 04° for 50'. The 
figures denoting the bearing may have been transposed in setting 
down, as 53 for 35. So, too, the chain may have been wrongly read, as 
48 for 52, the links having been counted from the wrong end. Or they 
may have counted from the wrong tag, as 48 for 38. Mistakes of the 
nature of these mentioned are common, so that in working from a 
bearing tree to find a corner, and not finding the stake at the place 
indicated in the notes, it will be well to test all these sources of error 
before giving up the search, for as I have said before, the post planted 
at the time of the oriymal survey is the best evidence of the corner it was 
intended to indicate. 

I next consider fences in their relations to corners. (Potts v Ever¬ 
hart, 26 Penn. St. Rep., 493.) Whether any particular fence may be 
depended on to indicate the true line will depend on the particular cir¬ 
cumstances attending that case. In a general and rough way, a fence 
will indicate to the surveyor where to begin looking for his corner. 
But the practice has been, and still is common, for the first settlers on 
a section to clear and fence beyond the line in order to have a clear 
place on which to set their permanent fence when they get ready to 


302 


A MANUAL OF LAND SURVEYING. 


build it. Afterward they forget where the line is and set the new fence 
where the old one stood. Many fences, too, were set without any sur¬ 
vey or any accurate knowledge where the line was and left there to 
await a convenient time to have the line established. So, too, where 
the land has been long settled and occupied, it is a common custom for 
adjoining land owners by consent to set the fence on one side of the 
true line, there to remain until they are ready to rebuild, the one 
party to have the use of the land for that time in consideration of clear¬ 
ing out and subduing the old fence row. The original parties fre¬ 
quently sell out or die, and the new owners have no knowledge of the 
agreement and suppose the fence to be on the true line. For these 
reasons, fences should be looked on with suspicion, unless corroborated 
by other evidence, and the surveyor should enquire pretty closely into 
the history of a fence before placing any great reliance on it to deter¬ 
mine the position of a corner. It may be the best of evidence, or it 
may be utterly worthless. 

It not unfrequently happens that there are no trustworthy marks 
near a corner to direct the surveyor in his search for the post or from 
which to replace it if it be destroyed. In these cases, he must visit the 
nearest corners he can find in each direction (varying with the circum¬ 
stances whether it be section corner or quarter post he wishes to find 
or restore), go through the process of identification with each of them, 
and then make his point so that it will bear the same relation to these 
corners as did the original corner post. Many very intelligent gentle¬ 
men suppose that if the surveyor can but find one of the corners of 
the original United States survey he can readily determine the position 
of all the rest from it. They were never more mistaken in their lives. 
The continual change in the direction of the magnetic needle, the un¬ 
certainty as to what its direction was when any particular line was 
run, the difference in the lengths of chains, and the difference in the 
men who use them, introduce so many elements of uncertainty into 
the operation as to render it one of little value, and not to be resorted 
to except in the absence of trustworthy evidence nearer at hand. 

If it be a section corner you desire to find or replace, and have ad¬ 
jacent quarter posts in each direction to work from, you will not be 
likely on the one hand to fall more than a rod or two out of the way, 
and on the other hand will not be likely to come within a foot or two 
of the right place. This method will assist you in seacliing for the 
original stake, and if that be destroyed, and no better evidence pre¬ 
sents itself, may be used to determine the point where the corner 
stake shall be placed. The chief difficulty in applying this method to 
determine corners arises from the fact that the measurements made 
on the original surveys were not uniform in length on different sec¬ 
tions, and frequently not on different parts of the same section. I have 
measured sections 22 and 211 on a level prairie, along the line of high- 


RE-LOCATION OF LOST CORNERS. 


303 


ways, where no obstacles of any kind interfered to prevent accurate 
work. I took the greatest possible care in the chaining to have it as 
accurate as chain work can be done. On the north line of section 22 
my chaining tallied exactly with that of the United States survey, viz., 
79.C0. On the north line of section 23, my measure was 80.9G, that of 
the United States survey, 80.40— a difference of 5G links. Fortunately, 
all the corners of the original survey on this two miles of line were 
well preserved, and the distance between quarter post and section cor¬ 
ners was uniform on the same section in both sections. But suppose 
that a part of them had been lost, and it was required to restore the 
middle section corner (n. e. of 22) from the remaining ones. Omit all 
consideration of corners, north or south, and there remain four differ¬ 
ent solutions of the problem, depending on which corners were lost 
and which preserved. Of these different solutions, one would place 
the corner 9% links, one 14 links, one 18% links, and one 28 links, all 
east of the true corner. This is not by any means an extreme instance, 
as I have observed discrepancies twice as great. It is given simply to 
show how unreliable is the evidence drawn from distant corners of the 
United States survey. 

Lastly, I shall consider the evidence of living persons. [Weaver v. 
Robinett, 17 Mo., 459; Chapman v. Twitchell, 37 Maine, 59; Dagget v. 
AViley, G Florida, 482: Lewen v. Smith, 7 Port. (Ala ),428; McCoy v. Gal¬ 
loway, 3 Han. (Ohio), 283; and Stover v. Freeman, 6 Mass., 441] Con¬ 
ceding all men to be equally honest in their evidence, there is a vast 
deal of difference among them with regard to tlieir habits of observa¬ 
tion and their ability to determine localities. Some have an exceed¬ 
ingly acute sense of locality, if we may so call it, and can determine 
very accurately the position of any object which they have been accus¬ 
tomed to see; while others seem to have little or no capacity of that 
sort. I have found many men who would describe accurately the sort 
of monument used to perpetuate a corner, and who would tell you that 
they could put their foot on the very spot to look for it; but when the 
trial came I have found but few of them who could locate the point 
within several feet, unless they had some object near at hand to assist 
the memory, and even then they would frequently fail. 

It may happen where a corner post lias been destroyed, that its loca¬ 
tion can be more nearly determined by the testimony of persons who 
were familiar with it when standing and can testify to its relations to 
other objects in its vicinity, than in any other way. But the surveyor 
in receiving this testimony should ascertain as far as possible what are 
the habits of accurate observation and the memory of localities pos¬ 
sessed by the person testifying, in order to know how much weight to 
give his testimony.” 


304 


A MANUAL OF LAND SURVEYING. 


CHAPTER XII. 

MISCELLANEOUS. 

1. Questions of Practice.—Answers to most if not 

all questions which arise in the surveyor’s practice will 
be found in the Supreme Court decisions which have 
been quoted. The following questions which have been 
raised in several surveyors’ associations, are given with 
the answers adopted in each case, or a reference to the 
law decision or principle which governs it. 

1. An interior section has its quarter posts out of line 
and not at equidistant points between the section corners. 
How shall the centre be determined? 

Am. At the intersection of straight lines from each 
quarter section corner to its opposite corresponding cor¬ 
ner. 

See page is2. Sec. 100, Second. 

2. IIow shall the quarter posts on the north and west 
lines of the township which were not established by the 
U. S, survey be located ? 

Ans. The corners of half and quarter sections, not 
marked on the surveys, shall be placed as nearly as possi¬ 
ble equidistant from those two corners which stand on 
the same line. 

See page 182, Sec. loo, First. 

Section 6 is an exception to this rule. 

See page 235. 

3. Posts for lines closing on the north and west boun¬ 
daries of townships are often off the boundary line to one 
side or the other. Shall the boundary line be deflected to 
pass through these posts ? 


MISCELLANEOUS. 


305 


Aiis. No. The posts serve to show the position of the 
section line, but the line itself stops at the township 
boundary.* 

Mich. Surv. Rep., 1881. 

4. Are the station or line trees marked on the govern¬ 
ment surveys and returned in the field notes, monu¬ 
ments of the lines? 

Ans. Yes. 

See page 182, Sec. loo, Second. 

Billingsley v. Bates, 30 Ala. 378. 

5. How shall the east and west quarter line of section 
30 be located, there having been no quarter post set on the 
east side of the section by the U. S. survey, because of a 
lake? 

Ans. Locate the west quarter post as directed in the 
answer to question 2. Then run the quarter line east on 
a course which is intermediate between the courses of the 
north and the south lines of the section. 

See page 232. 

6. A closing corner on the north or west boundary of 
the township is lost. The field notes do not give the dis¬ 
tance between the closing corner and the adjacent corner 
on the boundary. How shall it be restored ? 

Ans. Prolong the known portion of the line to its inter¬ 
section with the boundary and there set the corner. 

See Billingsley v. Bates, 30 Ala. 378; see p. 282- 


* Note.— The author has not met with any judicial determination of 
this question. Some very able surveyors hold a different view from 
that expressed above. But suppose that the deputy surveyor, not 
finding the standard corner, as frequently happens, ran his line 
directly over it, and planted his closing corner in the section line be¬ 
yond. It would then be impossible to deflect the township line so as 
to pass through both corners. 

It would seem to be a safe way for the surveyor, in making a survey 
on a section, to locate his lines with reference to the corners estab¬ 
lished for that section only; and leave any question of title, raised by 
overlapping or non-closing lines, to be settled by the courts. 


21 




306 A MANUAL OF LAND SURVEYING. 

7. Should section lines running north and south he run 
in a straight line between known corners to locate lost 
corners on interior sections ? 

Am. Not unless the original lines were actually straight 
lines between the known points, which they seldom are. 

See Moreland v. Page, 2 Clarkes, Iowa, 139; see p. 283 i 

Martz v. Williams, 67 Ill., 306; see p. 283. 

8. How shall the half-quarter corner on the quarter line 
be located on those quarter sections which adjoin the 
north and west lines of the township ? 

Am. Measure the distance from the centre of the sec¬ 
tion to the quarter post on the township line. 

Then place the corner on the quarter line at a distance 
of twenty chains proportionate measurement from the 
centre of the section. In order to prorate the distance, 
your own measure should be compared with a distance 
which is a mean between the distances given in the field 
notes as the length of the corresponding lines of the sec¬ 
tion on either side. For example, on section 3 the dis¬ 
tance by U. S. survey from the east l post to township 
line is 42.18; from the west £ post to township line is 
43.20; which gives a mean distance of 42.69. 

Commissioner McFarland gives the following reply to 
a similar question • 

Department of the Interior, j 
General Land Office, t 
Washington, D. C., February 11, 1882.) 

Isaac Teller, Esq., Webberville, Ingham County, Michigan: 

Sir—I am in receipt of your letter of the 5tli instant requesting in¬ 
formation in regard to the proper method of locating the quarter-quar¬ 
ter corners north of the legal centres of the northern tier of sections 
in a township when the present measurement of the east and west 
boundaries of the section differs from the original measurement. 

In reply, I have to state that the length of the quarter line from the 
south quarter corner to the township line is to be considered as the 
mean of the east and west boundaries of the section as given in the 
field notes, and where the present measurement of the section lines 
differs from the original measurement, the rule of proportionate 
measurement applies to the quarter line as well as to the section lines 
in the establishment of quarter-quarter corners on the half mile closing 


MISCELLANEOUS. 307 

jn the township boundary. See enclosed circular dated November 1, 
1879. 

The mean width of the north half of the section in the case stated 
by you is 40.18 chains, while by your chaining it is 42.42 chains (calling 
the distance to the east and west quarter line 40.00 chains), therefore 
the proportion will be as 40.18 : 42.42 :: 20.00 : 21.11 chains, the distance 
north of the centre of the section at which by your chaining the quar¬ 
ter-quarter corner should be located. 

Very respectfully, 

N. C. McFARLAND, Commissioner. 

9. In surveying sections fractional on the township line 
to restore lost quarter section corners, should the lines he 
divided pro rata according to the U. S. field notes, or 
should the south or east quarters be made full and the 
entire excess or deficiency be thrown into the fraction? 

Ans. Any difference between your measure and the 
government measure must be distributed proportionally 
between the different parts of the section. 

See p. 240, Sec. 100, Second. 

Moreland v. Page, 2 Clarkes, Iowa 139. 

Jones v. Kimble, 19 Wis. 429. 

Martz v. Williams, 67 Ill. 306, 

In Missouri, the Supreme Court holds (Knight v. Elliott, 57 Mo. 317) 
a different view, viz., that the difference in measure is all to be thrown 
into the fraction. 

It is difficult to see upon what grounds this decision can be upheld 
in view of the fact that all rights to the land were acquired and he’d 
under the law of Congress, which expressly states that the length of 
such lines as returned by the surveyor-general shall be held and con¬ 
sidered as the true length thereof. 

10. The accompanying figure 
is a copy of the plat of the U. S. 
survey of this quarter section. 

A owns the whole quarter, 
lie sells to B the W. \ of the N. 
W. \ of section 18, containing 
91_54_ acr es. At about the same 
time he sells to C the E. ^ ot 
the N. W. i of section 18, con¬ 
taining 91y 5 o 4 o acres. 


Northwest Quarter, Sec. 18. 


2S. 

2 o.o o 

A. / 03 . C 8 

J 80 . 


Fig. 72. 




308 


A MANUAL OF LAND SURVEYING. 


Where shall the surveyor run the dividing- line between 
B and C? 

Ans. The language of the deed clearly shows the inten¬ 
tion of A to sell and of B and C to purchase each the half 
of the area of the quarter section. The surveyor should 
so locate the line as to carry out the evident intent of 
the parties. See rule 2, p. 244 and rule 14, p. 284. The fact 
that the quarter is differently subdivided on the govern¬ 
ment plat has no bearing on this case. 

11. Certain early surveyed townships had three sets of 
corners on the range lines. (1) Those set when the range 
lines were run; (2) Those set as closing corners running 
east; (3) Those set as closing corners running west. What 
use is made of each set of corners ? 

Ans. The first corners set determine the location of the 
range line. The second and third sets of corners deter¬ 
mine the location of their respective section lines which 
close on and terminate at the range line. 



12. This figure shows a fractional township on the 
Ohio River. The figures show the dimensions of section 
1, as shown by the field notes of the United States survey. 
By a subsequent measure, 

AB — 82.25 chains, and AT) — 79.50 chains. 









MISCELLANEOUS. 


309 


IIow shall the northeast quarter of section 1 be laid 
off, no quarter-posts having been planted ? 

Ajis. Place the quarter-section corners on the north and 
east sides of the section in line with and midway between 
their respective section corners. Make the east and w r est 
quarter-line parallel with the south line of the section, 
placing the west quarter-post at the point where the 
quarter-line thus run intersects the section line. From 
the north quarter-post run the quarter-line south on a 
course which is a mean between the courses of the east 
and the west lines of the section, placing the south quar¬ 
ter post at the intersection of the section and quarter- 
section lines. 

The exceptional features of this case are that no quar¬ 
ter-posts were set on the United States survey, and that 
the east line of the section is just 80 chains in length, 
having been run from the north to the south. 



13. The description in the deed runs: “ Beginning at a 
stone (A ), at the N.W. corner of lot 401; thence east 112 ft. 
to a stone (/>); thence S. 3fH° W. 100 ft.; thence west par¬ 
allel with All to the west line of said lot 401; thence north 
on west line of said lot, 66 ft., to the place of beginning.” 
The points A and 11, and angle ABC , are fixed. F, by 














310 


A MANUAL OF LAND SURVEYING. 


construction and in fact, is 80 T Vo ft. distant, at right am 
gles from the line AB. 

1. Shall I locate CD parallel with AB, or locate D 66 ft. 
from A ? 

2. Have I any right to consider any apparent intention 
to locate 66 ft. or 80 T W ft. from A ? 

3. Have I, if I know it, any authority to consider the 
actual intention of the grantor to locate CD ? 

4. If the distance AB should actually measure 114 ft. 
am T to use it, or shall I make B 112 ft. from A ? 

A ns. 1. The answer to this question will depend upon 
the state of facts brought out in answer to questions 2 
and 3. If there he evidence showing what the intention 
and understanding of the parties to the conveyance was 
as to which of the two lines should be taken, that evi¬ 
dence would settle the question. If not, that construc¬ 
tion may be given to the deed which will operate most 
strongly against the grantor and give the grantee the 
greater amount of land. So far as anything is shown in 
the question, the deeds to the adjacent land might fur¬ 
nish the necessary evidence. 

2 and 3. Yes. Judge Cooley says, (see “Judicial Func¬ 
tions of Surveyors”): “The surveyor must inquire into 
all the facts, giving due prominence to the acts of parties 
concerned, and always keeping in mind * * * * that courts 
and juries may be required to follow after the surveyor 
over the same ground, and that it is exceedingly desirable 
that he govern his action by the same lights and the same 
rules that will govern theirs.” 

4. The monument controls the distance. 

14. A piece of land is sold, and described as commencing 
at the north quarter-post of section 15, and running 
thence east 100 rods; thence south 160 rods; thence west 
100 rods; thence north 160 rods, to the place of beginning: 
containing 100 acres, according to the United States sur¬ 
vey. 


MISCELLANEOUS. 


311 


Ques. How shall it be set off? 

Ans. The deed clearly indicates the understanding of 
the parties to the conveyance to be that the land should 
pass according to the rules that govern the United States 
survey. One of these rules is, that “the length of the 
boundary lines as returned by the surveyor-general shall 
be held and considered as the true length thereof.” Hence 
in this case, measure east from the quarter-post along the 
section line 25 chains of just such measure as the United 
States surveyors gave ; or in other words, of pro rata 
measurement. Suppose the distance by the field notes to 
be 40.32 chains from quarter-post to section corner. Then 

25.00 

lay off-— of that distance. Proceed in a similar man- 

40.32 

ner, running east on the quarter-line from the center of 
the section, and the two points thus located will be the 
corners of the 100 acres. To get the length of the south 
line of the H. E. I of the section by the United States 
survey, take the half sum of the measure given on the 
north and the south lines of the section. Supposing it to 
be 40.32 on the north, and 40.18 on the south, then the 
distance on the quarter-line would be equal to 


40.32 + 40.18 

-= 40.25, 

2 

25.00 

and you should measure off-of this distance for the 

• 40.25 


corner. 


2. The Rights, Duties and Responsibilities of 
Surveyors.—Surveyors, by the consent and acquiescence 
of the parties concerned, are usually the arbiters of dis¬ 
puted boundaries, and their decisions, when thus acqui¬ 
esced in by the parties, become in time as binding, and as 
much respected by the authorities, as the decisions of 
juries and courts of law. It is probable that at least 
ninety-nine per cent, of all questions of disputed bound- 





312 


A MANUAL OF LAND SURVEYING. 


aries are thus settled by the interested parties themselves, 
in accordance with the decision of the surveyor. 

Surveyors, from constantly exercising this seeming 
authority, come at last in many cases to believe it to be 
absolute and final, something which must be respected, 
overlooking the fact that the only force their decisions 
have comes from the consent of the parties. When that 
consent is withheld, the case goes to the courts for settle¬ 
ment; and thus the courts have in some cases felt called 
upon to define the surveyor’s standing before the law. 
They say: 

1. “Surveyors have no more authority than other men 
to determine boundaries, of their own motion. All bounds 
and starting points are questions of fact to be determined 
by testimony. Surveyors may or may not have in certain 
cases means of judgment not possessed by others, but the 
law can not and does not make them arbiters of private 
rights. 

Cronin v. Gore, 38 Mich. 381. 

2. The law recognizes surveyors as useful assistants in 
doing the mechanical work of measurement, and calcu¬ 
lation, and also allows such credit to their judgment as 
belongs to any experience which may give it value in 
cases where better means of information do not exist. 
But the determination of facts belongs exclusively to 
courts and juries. Where a section line or other starting 
point actually exists, is always a question of fact, and 
cannot be left to the opinion of an expert for final decis¬ 
ion. And where, as is generally the case in an old com¬ 
munity, boundaries have been fixed by long use and 
acquiescence, it would be contrary to all reason to have 
them interfered with on any abstract notion of science. 

Stewart v. Carleton, 31 Mich. 273. 

Gregory v. Knight, 50 Mich. Gl. 

3. New surveys disturbing old boundaries are not to be 
encouraged. 

Toby v. Secor, Wisconsin. N. W. Reporter, Vol. 19, p. 79. 


MISCELLANEOUS. 


313 


4. Lines long unquestioned ought not to be disturbed 
upon a mere disagreement among surveyors, especially 
when the last survey is made under the unfavorable cir¬ 
cumstances of corner posts and witness trees being gone, 
which it is probable to suppose were in existence at the 
time of the first survey. 

Case v. Trapp, 49 Mich. 59. 

5. County surveyors’ certificate are not admissible in 
evidence unless they contain all the particulars required 
by the statute to be entered in the surveyor’s record. 

Smith v. Rich, 37 Mich. 549. 

The statute of Michigan required the length of all lines 
run, the area of lands surveyed, and other particulars, to 
be entered in the county surveyor’s record. In the above 
case the survey was solely to find the location of a corner 
post. As the surveyor’s certificate did not show any area 
of land surveyed, it was not admitted in evidence. 

6. A surveyor was called on to survey the line of a 
highway. He performed the work so unskillfully as to 
render a new survey necessary. A large amount of road 
constructed at great expense, on the line designated by 
the surveyor before the mistake was discovered, had to 
be abandoned. Action was brought to recover damages. 
Held, that whether the defendant was a professional or 
official surveyor,or represented himself as such, his under¬ 
taking was that he should bring to the work the neces¬ 
sary knowledge and skill to perform the same properly 
and correctly; and if he failed to do so, and the plaintiff 
suffered damage in consequence of such failure, the plain¬ 
tiff will be entitled to recover. 

Commissioner of Highways v. Beebe, Midi. Sup. Court. 

N. W. Rep., Vol. 20, No. 16, 

The following paper, by Chief Justice Cooley, of the 
Supreme Court of Michigan, discusses more fully the 
surveyor’s functions; 


314 


A MANUAL OF LAND SURVEYING. 


3 • The Judicial Functions of Surveyors.— 

When a man has had a training in one of the exact 
sciences, where every problem within its purview is sup¬ 
posed to be susceptible of accurate solution, he is likely 
to be not a little impatient when he is told that, under 
some circumstances, he must recognize inaccuracies, and 
govern his action by facts which lead him away from the 
results which theoretically he ought to reach. Observa¬ 
tion warrants us in saying that this remark may fre¬ 
quently be made of surveyors. 

In the State of Michigan, all our lands are supposed to 
have been surveyed once or more, and permanent monu¬ 
ments fixed to determine the boundaries of those who 
should become proprietors. The United States, as orig¬ 
inal owner, caused them all to be surveyed once by sworn 

t 

officers, and as the plan of subdivision was simple, and 
was uniform over a large extent of territory, there should 
have been, with due care, few or no mistakes; and long 
rows of monuments should have been perfect guides to 
the place of any one that chanced to be missing. The 
truth unfortunately is, that the lines were very carelessly 
run, the monuments inaccurately placed; and, as there- 
corded witnesses to these were many times wanting in 
permanency, it is often the case that w T hen the monument 
was not correctly placed^ it is impossible to determine by 
the record, by the aid of anything on the ground, where 
it was located. The incorrect record of course becomes 
worse than useless when the witnesses it refers to have 
disappeared. 

It is, perhaps, generally supposed that our town plats 
were more accurately • surveyed, as indeed they should 
have been, for in general there can have been no difficulty 
in making them sufficiently perfect for all practical pur¬ 
poses. Many of them, however, were laid out in the 
woods; some of them by proprietors themselves, without 
either chain or compass, and some by imperfectly trained 
surveyors, who, when land was cheap, did not appreciate 


MISCELLANEOUS. 


315 


the importance of having- correct lines to determine 
boundaries when land should become dear. The fact 
probably is, that town surveys are quite as inaccurate as 
those made under authority of the general government. 

It is now upwards of fifty years since a major part of 
the public surveys in what is now the State of Michigan 
were made under authority of the United States. Of the 
lands south of Lansing, it is now forty years since the 
major part were sold, and the work of improvement be¬ 
gan. A generation has passed away since they were con¬ 
verted into cultivated farms, and few if any of the 
original corner and quarter stakes now remain. 

The corner and quarter stakes were often nothing but 
green sticks driven into the ground. Stones might be 
put around or over these if they were handy, but often 
they were not, and the witness trees must be relied upon 
after the stake was gone. Too often the first settlers were 
careless in fixing their lines with accuracy while monu¬ 
ments remained, and an irregular brush fence, or some¬ 
thing equally untrustworthy, may have been relied upon 
to keep in mind where the blazed line once was. A fire 
running through this might sweep it away, and if nothing 
was substituted in its place, the adjoining proprietors 
might in a few years be found disputing over their lines, 
and perhaps rushing into litigation, as soon as they had 
occasion to cultivate the land along the boundary. 

If now the disputing parties call in a surveyor, it is not 
likely that any one summoned would doubt or question 
that his duty was to find, if possible, the place of the 
original stakes which determined the boundary line be¬ 
tween the proprietors. However erroneous may have 
been the original survey, the monuments that were set 
must nevertheless govern, even though the effect be to 
make one half-quarter section ninety acres and the one 
adjoining seventy; for parties buy, or are supposed to 
buy, in reference to these monuments, and are entitled to 
what is within their lines, and no more, be it more or less. 


A MANUAL OF LAND SURVEYING. 


316 

While the witness trees remain, there can generally be no 
difficulty in determining the locality of the stakes. 

When the witness-trees are gone, so that there is no 
longer record evidence of the monuments, it is remark¬ 
able how many there are who mistake altogether the duty 
that now devolves upon the surveyor. It is by no means 
uncommon that we find men, whose theoretical education 
is thought to make them experts, who think that when 
the monuments are gone, the only thing to be done is to 
place new monuments where the old ones should have 
been, and would have been, if placed correctly. This is a 
serious mistake. The problem is now the same that it 
was before: To ascertain by the best lights of which the 
case admits, where the original lines were. The mistake 
above alluded to, is supposed to have found expression in 
our legislation; though it is possible that the real intent 
of the act to which we shall refer is not what is com¬ 
monly supposed. 

An act passed in 1869, (Compiled Laws, § 593), amending 
the laws respecting the duties and powers of county sur¬ 
veyors, after providing for the case of corners which can 
be identified by the original field notes or other, unques¬ 
tionable testimony, directs as follows: 

“ Second. Extinct interior section corners must be re-established at 
the intersection of two right lines joining the nearest known points on 
the original section lines east and west and north and south of it. 

“ Third. Any extinct quarter-section corner, except on fractional 
lines, must be re-established equidistant and in a right line between 
the section corners; in all other cases at its proportionate distance 
between the nearest original corners on the same line.” 

The corners thus determined, the surveyors are required 
to perpetuate by noting bearing trees when timber is near. 

To estimate properly this legislation, we must start with 
the admitted and unquestionable fact that each purchaser 
from government bought such land as was within the 
original boundaries, and unquestionably owned it up to 
the time when the monuments became extinct. If the 
monument was set for an interior section corner, but did 


MISCELLANEOUS. 


317 


4 

not happen to be “ at the intersection of two right lines 
joining the nearest known points on the original section 
lines east and west and north and south of it,” it never¬ 
theless determined the extent of his possessions, ana he 
gained or lost according as the mistake did or did not 
favor him. 

It will probably be admitted that no man loses title to 
his land or any part thereof merely because the evidences 
become lost or uncertain. It may become more difficult 
for him to establish it as against an adverse claimant, 
but theoretically the right remains; and it remains as a 
potential fact so long as he can present better evidence 
than any other person. And it may often happen that 
notwithstanding the loss of all trace of a section corner 
or quarter stake, there will still be evidence from which 
any surveyor will be able to determine with almost abso¬ 
lute certainty where the original boundary was between 
the government subdivisions. 

There are two senses in which the word extinct may be 
used in this connection: One, the sense of physical dis¬ 
appearance ; the other, the sense of loss of all reliable 
evidence. If the statute speaks of extinct corners in the 
former sense, it is plain that a serious mistake was made 
in supposing that surveyors could be clothed with author¬ 
ity to establish new corners by an arbitrary rule in such 
cases. As well might the statute declare that if a man 
loses his deed, he shall lose his land altogether. 

But if by extinct corner is meant one in respect to the 
actual location of which all reliable evidence is lost, then 
the following remarks are pertinent : 

1. There would undoubtedly be a presumption in such 
a case that the corner was correctly fixed by the govern¬ 
ment surveyor where the field notes indicated it to be. 

2. But this is only a presumption, and may be over¬ 
come by any satisfactory evidence showing that in fact 
it was placed elsewhere. 


318 A MANUAL OF LAND SURVEYING. 

3.. No statute can confer upon a county surveyor the 
power to “ establish ” corners, and thereby bind the par¬ 
ties concerned. Nor is this a question merely of conflict 
between State and federal law; it is a question of prop¬ 
erty right. The original surveys must govern, and the 
laws under which they were made must govern, because 
the land was bought in reference to them; and any legis¬ 
lation, whether State or federal, that should have the 
effect to change these, would be inoperative, because dis¬ 
turbing vested rights. 

4. In any case of disputed lines, unless the parties 
concerned settle the controversy by agreement, the deter¬ 
mination of it is necessarily a judicial act, and it must 
proceed upon evidence, and give full opportunity for a 
hearing. No arbitrary rules of survey or of evidence can 
be laid down whereby it can be adjudged. 

The general duty of a surveyor in such a case is plain 
enough. He is not to assume that a monument is lost 
until after he has thoroughly sifted the evidence and 
found himself unable to trace it. Even then he should 
hesitate long before doing anything to the disturbance of 
settled possessions. Occupation, especially if long con¬ 
tinued, often affords very satisfactory evidence of the 
original boundary when no other is attainable; and the 
surveyor should inquire when it originated, how, and 
why the lines were then located as they were, and whether 
a claim of title has always accompanied the possession, 
and give all the facts due force as evidence. Unfortun¬ 
ately, it is known that surveyors sometimes, in supposed 
obedience to the State statute, disregard all evidences of 
occupation and claim of title, and plunge whole neigh¬ 
borhoods into quarrels and litigation by assuming to 
“ establish ” corners at points with which the previous 
occupation cannot harmonize. It is often the case that 
where one or more corners are found to be extinct, all 
parties concerned have acquiesced in lines which were 
traced by the guidance of some other corner or landmark, 


MISCELLANEOUS. 


319 


which may or may not have been trustworthy; but to 
bring’ these lines into discredit when the people concerned 
do not question them, not only breeds trouble in the 
neighborhood, but it must often subject the surveyor 
himself to annoyance and perhaps discredit, since in a 
legal controversy the law as well as common sense must 
declare that a supposed boundary line long acquiesced in 
is better evidence of where the real line should be than 
any survey made after the original monuments have dis¬ 
appeared. (Stewart v. Carleton, 31 Mich. Beports, 270; 
Diehl v. Zanger, 39 Mich. Beports, 001.) And county sur¬ 
veyors, no more than any others, can conclude parties by 
their surveys. 

The mischiefs of overlooking the facts of possession 
most often appear in cities and villages. In towns the 
block and lot stakes soon disappear; there are no witness 
trees, and no monuments to govern except such as have 
been put in their places, or where their places were sup¬ 
posed to be. The streets are likely to be soon marked off 
by fences, and the lots in a block will be measured off 
from these, without looking farther. Now it may per¬ 
haps be known in a particular case that a certain monu¬ 
ment still remaining was the starting point in the original 
survey of the town plat; or a surveyor settling in the 
town may take some central point as the point of depart¬ 
ure in his surveys, and assuming the original plat to be 
accurate, he will then undertake to find all streets and 
all lots by course and distance according to the plat, 
measuring and estimating from his point of departure. 
This procedure might unsettle every line and every mon¬ 
ument existing by acquiescence in the town; it would be 
very likely to change the lines of streets, and raise con¬ 
troversies everywhere. Yet this is what is sometimes 
done; the surveyor himself being the first person to raise 
the disturbing questions. 

Suppose, for example, a particular village street has 
been located by acquiescence and used for many years, 



320 A MANUAL OF LAND SURVEYING. 

and the proprietors in a certain block have laid off their 
lots in reference to this practical location. Two lot own¬ 
ers quarrel, and one of them calls in a surveyor, that he 
may make sure his neighbor shall not get an inch of land 
from him. This surveyor undertakes to make his survey 
accurate, whether the original was so or not, and the first 
result is, he notifies the lot owners that there is error in 
the street line, and that all fences should be moved, say 
one foot to the east. Perhaps he goes on to drive stakes 
through the block according to this conclusion. Of 
course, if he is right in doing this, all lines in the village 
will be unsettled; but we will limit our attention to the 
single block. It is not likely that the lot owners gener¬ 
ally will allow the new survey to unsettle their posses¬ 
sions, but there is always a probability of finding some 
one disposed to do so. We shall then have a lawsuit; 
and with what result ? 

It is a common error that lines do not become fixed by 
acquiescence in a less time than twenty years. In fact, 
by statute, road lines may become conclusively fixed in 
ten years; and there is no particular time that shall be 
required to conclude private owners, where it appears 
that they have accepted a particular line as their bound¬ 
ary, and all concerned have cultivated and claimed up to 
it. Public policy requires that such lines be not lightly 
disturbed, or disturbed at all after the lapse of any con¬ 
siderable time. The litigant, thererore, who in such a 
case pins his faith on the surveyor, is likely to suffer for 
his reliance, and the surveyor himself to be mortified by 
a result that seems to impeach his judgment. 

Of course nothing in what has been said can require a 
surveyor to conceal his own judgment, or to report the 
facts one way when he believes them to be another. He 
has no right to mislead, and he may rightfully express 
his opinion that an original monument was at one place, 
when at the same time he is satisfied that acquies¬ 
cence has fixed the rights of parties as if it were at an- 


MISCELLANEOUS. 


321 


other. But he would do mischief if he were to attempt 
to “establish ” monuments which he knew would tend 
to disturb settled rights; the farthest he has a right to 
go, as an officer of the law, is to express his opinion where 
the monument should be, at the same time that he im¬ 
parts the information to those who employ him, and who 
mignt otherwise be misled, that the same authority that 
makes him an officer and entrusts him to make surveys, 
also allows parties to settle their own boundary lines, and 
considers acquiescence in a particular line or monument, 
for any considerable period, as strong if not conclusive 
evidence of such settlement. The peace of the commu¬ 
nity absolutely requires this rule. It is not long since, 
that in one of the leading cities of the State an attempt 
was made to move houses two or three rods into a street, 
on the ground that a survey under which the street had 
been located for many years, had been found on a more 
recent survey to be erroneous. 

From the foregoing, it will appear that the duty of the 
surveyor where boundaries are in dispute must be varied 
by the circumstances. 1. He is to search for original 
monuments, or for the places where they were originally 
located, and allow these to control if he finds them, unless 
he has reason to believe that agreements of the parties, 
express or implied, have rendered them unimportant. By 
monuments in the case of government surveys we mean 
of course the corner and quarter-stakes; blazed lines or 
marked trees on the lines are not monuments: they are 
merely guides or finger posts, if we may use the expres¬ 
sion, to inform us with more or less accuracy where the 
monuments may be found. 2. If the original monuments 
are no .longer discoverable, the question of location be¬ 
comes one of evidence merely. It is merely idle for any 
State statute to direct a surveyor to locate or “ establish ” 
a corner, as the place of the original monument, accord¬ 
ing to some inflexible rule. The surveyor, on the other 
hand, must inquire into all the facts: giving due promi- 




322 


A MANUAL OF LAND SURVEYING. 


nence to the acts of parties concerned, and always keep¬ 
ing in mind, first, that neither his opinion nor his survey 
can be conclusive upon parties concerned; and, second, 
that courts and juries may be required to follow after the 
surveyor over the same ground, and that it is exceedingly 
desirable that he govern his action by the same lights 
and the same rules that will govern theirs. 

It is always possible, when corners are extinct, that the 
surveyor may usefully act as a mediator between parties, 
and assist in preventing legal controversies by settling 
doubtful lines. Unless he is made for this purpose an 
arbitrator by legal submission, the parties, of course, even 
if they consent to follow his judgment, cannot, on the 
basis of mere consent, be compelled to do so; but if he 
brings about an agreement, and they carry it into effect 
by actually conforming their occupation to his lines, the 
action will conclude them. Of course, it is desirable that 
all such agreements be reduced to writing; but this is not 
absolutely indispensable if they are carried into effect 
without. 

Meander Lines.—The subject to which allusion will 
now be made, is taken up with some reluctance, because 
it is believed the general rules are familiar. Nevertheless, 
it is often found that surveyors misapprehend them, or 
err in their application; and as other interesting topics 
are somewhat connected with this, a little time devoted 
to it will probably not be altogether lost. The subject is 
that of meander lines. These are lines traced along the 
shores of lakes, ponds, and considerable rivers, as the 
measures of quantity when sections are made fractional 
by such waters. These have determined the price to be 
paid when government lands were bought, and perhaps 
the impression still lingers in some minds that the mean¬ 
der lines are boundary lines, and that all in front of them 
remains unsold. Of course this is erroneous. There was 
never any doubt that, except on the large navigable 
rivers, the boundary of the owners of the banks is the 


MISCELLANEOUS. 


323 


middle line of the river; and while some courts have held 
that this was the rule on all fresh-water streams, large 
and small, others have held to the doctrine that the title 
to the bed of the stream below low-water mark is in the 
State, while conceding to the owners of the banks all 
riparian rights. The practical difference is not very im¬ 
portant. In this State, the rule that the center line is the 
boundary line, is applied to all our great rivers, including 
the Detroit, varied somewhat by the circumstance of 
there being a distinct channel for navigation, in some 
cases, with the stream in the main shallow, and also 
sometimes by the existence of islands. 

The troublesome questions for surveyors present them¬ 
selves when the boundary line between two contiguous 
estates is to be continued from the meander line to the 
center line of the river. Of course, the original survey 
supposes that each purchaser of land on the stream has 
a water front of the length shown by the field notes; and 
it is presumable that he bought this particular land be¬ 
cause of that fact. In many cases it now happens that 
the meander line is left some distance from the shore by 
the gradual change of course of the stream, or diminu¬ 
tion of the flow of water. Now the dividing line be¬ 
tween two government subdivisions might strike the 
meander line at right angles, or obliquely; and, in some 
cases, if it were continued in the same direction to the 
center line of the river, might cut off from the water one 
of the subdivisions entirely, or at least cut it off from 
any privilege of navigation, or other valuable use of the 
water, while the other might have a water front much 
greater than the length of a line crossing it at right 
angles to its side lines. The effect might be that, of two 
government subdivisions of equal size and cost, one 
would be of very great value as water-front property, 
and the other comparatively valueless. A rule which 
would produce this result would not be just, and it has 
not been recognized in the law. 


324 


A MANUAL Or LAND SUIiYEYINO. 


Nevertheless it is not easy to determine what ought to 
be the correct rule for every case. If the river has a 
straight course, or one nearly so, every man’s equities 
will be preserved by this rule: Extend the line of divi¬ 
sion between the two parcels from the meander line to 
the center line of the river, as nearly as possible at right 
angles to the general course of the river at that point. 
This will preserve to each man the water front which the 
field notes indicated, except as changes in the water may 
have affected it, and the only inconvenience will be that 
the division line between different subdivisions is likely 
to be more or less deflected where it strikes the meander 
line. 

This is the legal rule, and is not limited to government 
surveys, but applies as well to water lots which appear as 
such on town plats. (Bay City Gas Light Co. v. The In¬ 
dustrial Works, 28 Mich. Reports, 182.) It often happens, 
therefore, that the lines of city lots bounded on naviga¬ 
ble streams are deflected as they strike the bank, or the 
line where the bank was when the town was first laid out. 

When the stream is very crooked, and especially if there 
are short bends, so that the foregoing rule is incapable of 
strict application, it is sometimes very difficult to deter- 
mine what shall be done; and in many cases the surveyor 
may be under the necessity of working out a rule for 
himself. Of course his action cannot be conclusive; but 
if he adopts one that follows as nearly as the circum¬ 
stances will admit, the general rule above indicated, so as 
to divide as near as may be the bed of the stream among 
the adjoining owners in proportion to their lines upon 
the shore, his division, being that of an expert, made upon 
the ground and with all available lights, is likely to be 
adopted as law for the case. Judicial decisions, into 
which the surveyor would find it prudent to look under 
such circumstances, will throw light upon his duties and 
may constitute a sufficient guide when peculiar cases 
arise. Each riparian lot owner ought to have a line on 


MISCELLANEOUS. 


325 


the legal boundary, namely, the center line of the stream 
proportioned to the length of his line on the shore and 
the problem in each case is, how this is to be given him. 
Alluvion, when a river imperceptibly changes its course, 
will be apportioned by the same rules. 

The existence of islands in a stream when the middle 
line constitutes a boundary, will not affect the apportion¬ 
ment unless the islands were surveyed out as government 
subdivisions in the original admeasurement. Wherever 
that was the case, the purchaser of the island divides the 
bed of the stream on each side with the owner of the 
bank, and his rights also extend above and below the 
solid ground, and are limited by the peculiarities of the 
bed and the channel. If an island was not surveyed as a 
government subdivision previous to the sale of the bank, 
it is of course impossible to do this for the purposes of 
government sale afterward, for the reason that the rights 
of the bank owners are fixed by their purchase; when 
making that they have a right to understand that all land 
between the meander lines, not separately surveyed and 
sold, will pass with the shore in the government sale: 
and having this right, anything which their purchase 
would include under it cannot afterward be taken from 
them. It is believed, however that the federal courts 
would not recognize the applicability of this rule to large 
navigable rivers, such as those uniting the great lakes. 

On all the little lakes of the state which are mere ex¬ 
pansions near their mouths of the rivers passing through 
them—such as the Muskegon, Pere Marquette and Manis¬ 
tee—the same rule of bed ownership has been judicially 
applied that is applied to the rivers themselves; and the 
division lines are extended under the water in the same 
way. (Rice v, Ruddiman, 10 Mich., 125.) If such a lake 
were circular, the lines would converge to the center; if 
oblong or irregular, there might be a line in the middle 
on which they would terminate, whose course would bear 
some relation to that of the shore. But it can seldom be 





326 


A MANUAL OF LAND SURVEYING. 


important to follow the division line very far under the 
water, since all private rights are subject to the public 
rights of navigation and other use, and any private use 
of the lands inconsistent with these would be a nuisance, 
and punishable as such. It is sometimes important, how¬ 
ever, to run the lines out for considerable distance, in 
order to determine where one may lawfully moor vessels 
or rafts, for the winter, or cut ice. The ice crop that 
forms over a man’s land of course belongs to him. (Lor- 
man v. Benson, 8 Mich., 18; People’s Ice Co. v. Steamer 
Excelsior, recently decided.) 

What is said above will show how unfounded is the 
notion, which is sometimes advanced, that a riparian 
proprietor on a meandered river may lawfully raise the 
water in the stream without liability to the proprietors 
above, provided he does not raise it so that it overflows 
the meander line. The real fact is that the meander line 
has nothing to do with such a case, and an action will lie 
whenever he sets back the water upon the proprietor 
above, whether the overflow be below the meander lines 
or above them. 

As regards the lakes and ponds of the state, one may 
easily raise questions that it would be impossible for him 
to settle. Let us suggest a few questions, some of which 
are easily answered, and some not: 

1. To whom belongs the land under these bodies of 
water, where they are not mere expansions of a stream 
flowing through them ? 

2. What public rights exist in them? 

3. If there are islands in them which were not sur¬ 
veyed out and sold by the United States, can this be done 
now ? 

Others will be suggested by the answers given to these. 

It seems obvious that the rules of private ownership 
which are applied to rivers cannot be applied to the great 
lakes. Perhaps it should be held that the boundary is at 
low water mark, but improvements beyond this would 


MISCELLANEOUS. 


327 


only become unlawful when they became nuisances 
Islands in the great lakes would belong to the United 
States until sold, and might be surveyed and measured 
for sale at any time. The right to take fish in the lakes, 
or to cut ice, is public like the right of navigation, but is 
to be exercised in such manner as not to interfere with 
the rights of shore owners. But so far as these public 
rights can be the subject of ownership, they belong to 
the state, not to the United States; and so, it is believed, 
does the bed of a lake also. (Pollord v. Hagan, 3 Howard’s 
U. S. Reports.) But such rights are not generally consid¬ 
ered proper subjects of sale, but like the right to make 
use of the public highways, they are held by the state in 
trust for all the people. 

What is said of the large lakes may perhaps be said also 
of many of the interior lakes of the state; such, for ex¬ 
ample, as Houghton, Higgins, Cheboygan, Burt’s, Mullet. 
Whitmore, and many others. But there are many little 
lakes or ponds which are gradually disappearing, and the 
shore proprietorship advances pari passu as the waters 
recede. If these are of any considerable size—say, even 
a mile across—there may be questions of conflicting 
rights which no adjudication hitherto made could settle. 
Let any surveyor, lor example, take the case of a pond of 
irregular form, occupying a mile square or more of terri¬ 
tory, and undertake to determine the rights of the shore 
proprietors to its bed when it shall totally disappear, and 
he will find he is in the midst of problems such as proba¬ 
bly he has never grappled with, or reflected upon before. 
But the general rules for the extension of shore lines, 
which have already been laid down, should govern such 
cases, or at least should serve as guides in their settle¬ 
ment. 

Where a pond is so small as to be included within the 
lines of a private purchase from the government, it is not 
believed the public have have any rights in it whatever. 
Where it is not so included, it is believed they have rights 


328 


A MANUAL OF LAND SURVEYING. 


of fishery, rights to take ice and water, and rights of nav¬ 
igation for business or pleasure. This is the common 
belief, and probably the just one. Shore rights must not 
be so exercised as to disturb these, and the states may 
pass all proper laws for their protection. It would be 
easy with suitable legislation to preserve these little 
bodies of water as permanent places of resort for the 
pleasure and recreation of the people, and there ought to 
be such legislation. 

If the state should be recognized as owner of the beds 
of these small lakes and ponds, it would not be owner for 
the purpose of selling. It would be owner only as trustee 
for the public use; and a sale would be inconsistent with 
the right of the bank owners to make use of the water in 
its natural condition in connection with their estates. 
Some of them might be made salable lands by draining; 
but the state could not drain, even for this purpose, 
against the will of the shore owners, unless their rights 
were appropriated and paid for. 

Upon many questions that might arise between the 
state as owner of the bed of a little lake and the shore 
owners, it would be presumptuous to express an opinion 
now, and fortunately the occasion does not require it. 

I have thus indicated a few of the questions with which 
surveyors may now and then have occasion to deal, and 
to which they should bring good sense and sound judg¬ 
ment. Surveyors are not and cannot be judicial officers, 
but in a great many cases they act in a quasi judicial 
capacity with the acquiescence of parties concerned; and 
it is important for them to know by what rules they are 
to be guided in the discharge of their judicial functions. 
What I have said cannot contribute much to their en¬ 
lightenment, but I trust will not be wholly without 
value. 


MAP DRAWING AND LETTERING. 


329 


CHAPTER XITT. 

MAP DRAWING AND LETTERING. 

BY C. R. DENISON. 

Materials.—Without entering too minutely into de¬ 
tail, let us briefly consider the materials to be used, their 
quality and adaptability to the purpose in hand; and, 
first, as to 

Paper .—The essential quality of a paper for map 
drawing is a hard, firm surface of uniform texture, which 
will take ink or water color smoothly and readily, and 
which is sufficiently tough to bear the use of instruments 
and rubber without sensible injury to its surface. Per¬ 
haps no paper fulfills these requirements better than 
Whatman’s. It comes in sheets of various sizes up to 
“antiquarian,” 31x53 inches. These papers are either hot 
or cold pressed, the former with a fine, smooth surface, 
particularly adapted for pen work; the latter has a 
somewhat coarse-grained surface, and is more especially 
designed for water color drawing, or plats on which the 
brush is to be used more or less. Of course there is a 
large variety of paper, and for various purposes, but 
within the limits of the size above given, it would be 
difficult to find anything superior to the product of 
“Whatman’s Turkey Mills,” which words can usually be 
read in water-line on each sheet. This paper also im¬ 
proves with age, and an old stock is consequently rather 
more valuable. 

Pens —Probably the best pens for mapping purposes 
are Gillott’s. His lithographic pen, mapping pen, and 
crow-quill are all valuable in free-hand work, but gen¬ 
erally, for ordinary use, his writing pen No. 303 will be 




330 


A MANUAL OF LAND SURVEYING. 


found very serviceable, while for filling in the bodies of 
letters of moderate size nothing is more useful than a 
common stub-pen, though some draughtsmen prefer a 
regular old goose-quill. 

Ink. —It is well to be somewhat critical in the matter 
of ink. There are a number of brands of liquid ink on 
the market, some of which are good, some admirable for 
special purposes, but after all no sufficient substitute can 
be found for the best quality of India ink, and when 
ground with a few drops of water, on a Keuffel & Esser 
ink slab, the time occupied is so slight that it is more 
than compensated by the satisfaction experienced in its 
use. Most emphatically India ink should be of a fine 
quality, and as a rule the higher the price the better 
the ink. True economy would pronounce in favor of 
an expensive cake, a fragment of which would last a 
draughtsman through the natural term of an active 
professional life. The best ink works up smoothly with 
water, forming a perfectly black mixture, and after dry¬ 
ing upon the paper forms a fine, glossy surface. For use 
it should be just black , as pale ink will make the boldest 
drawing look weak, and, on the other hand, if too thick it 
will constantly annoy the draughtsman by clogging the 
pen. After once grinding up a quantity of ink, a drop or 
two of pure glycerine will prevent its drying away for’ 
some time, and does not injure the mixture. If it is 
preferred, a liquid ink can be prepared by pulverizing a 
portion of the cake, and boiling it in a small quantity of 
water till the proper consistency is reached. It can then 
be kept in a vial and used as desired. Such a preparation, 
however, is apt to become offensive in the course of a 
week or two, and should then be re-boiled after the addi¬ 
tion of fresh water. A common test-tube is the best 
vessel in which to make the solution. After grinding ink 
from a cake of fine quality, it is well always to wipe it 
off carefully with a soft cloth, as this prevents waste by 
cracking and consequent disintegration of the cake. 

Instruments—As to instruments for map drawing—in 


MAP DRAWING AND LETTERING. 


331 


addition, of course, to drawing-board, T square and tri¬ 
angle—but few are needed, but these few should be of 
good, not to say of the best quality, and ought to consist 
in general of at least one large right-line pen, for draw¬ 
ing heavy lines, such as borders; one small right-line 
pen, for the finer lines; a pair of spacing dividers, and a 
large pair of dividers, with pen, pencil and needle-points. 
These instruments should be kept in perfect condition, or 
they cannot be made to do good work. The pens should 
be frequently sharpened by gently passing the outer sur¬ 
faces of the blades over a fine-grained stone, used dry. 
A fine knife-edged stone is sometimes applied to the 
inner surfaces, but this must be done with great care. 
When properly sharpened the points will be exactly even, 
and sharp enough to make, with slight pressure, a fine 
cut on the finger nail. A magnifying glass will aid in 
determining when the points are smooth and even. The 
best test, however, is to fill the pen and draw a series of 
lines, when, if they are all sharp and clear and unbroken, 
the pen is in good condition. The best implement for 
charging a drawing pen with ink is a common quill tooth¬ 
pick, using the long pointed end. 

Execution .—Having touched briefly upon the materials 
used by the draughtsman in his work, let us now con¬ 
sider, somewhat more carefully, the manner and methods 
of execution. While no one will gainsay the fact that 
the first and supreme requirement in a map is accuracy, 
or correctness, that quality of reliability without which 
it is not what it pretends to be — a map—nevertheless 
there are other qualities which it is by no means wise or 
judicious to neglect. Among these may be mentioned 
clearness, precision, legibility, neatness and sharpness of 
execution, and a certain prepossessing appearance which 
inspires the observer with confidence in the skill and 
powers of the maker. In fact, no map or technical 
drawing is above suspicion, or safe from the shadow of 
a doubt, when it bears upon its face traces of weakness 
which mark a want of knowledge and ability even in so 


332 


A MANUAL OF LAND SURVEYING. 


simple a matter as, for instance, the form and proportions 
of the letters of the alphabet. 

Furthermore, as a map or plan addresses itself directly 
to the eye as a product of skilled labor in one department 
of the art of design, it is not only right and proper, but 
essential to its complete success, that it should produce 
a pleasing impression, satisfying the demands of good 
taste, which can only be the result of the proper applica¬ 
tion of the laws of design. 

Principles .—We may pretty safely assume that the 
demands of a cultivated taste will be met by a reason¬ 
ably close adherence to the following simple principles: 
First —Construction may be decorated, but decoration 
should never be purposely and obviously the object of 
such construction. Second — Generally ornamentation 
should be based upon geometrical construction, and 
should make no attempt at relief or pictorial represen¬ 
tation, and “natural objects should not be used as orna¬ 
ment, but conventional representations founded upon 
them, suiliciently suggestive to convey the intended image 
to the mind without destroying the unity of the object 
they are employed to decorate.” Third—“ Harmony of 
form consists in the proper balancing and contrast of the 
straight, the inclined and the curved.” These are funda¬ 
mental propositions from one of the highest authorities 
on decorative design. While these principles are safe 
guides, and doubtless there are powerful motives for the 
introduction and use of geometrical forms for decora¬ 
tive purposes, founded upon their inherent beauty and 
almost universal employment, still, if any authority can 
be claimed for precedent, custom seems to have sanc¬ 
tioned the use for map decoration, to a limited extent, of 
vignettes, free-hand sketches, and symbolic devices, as 
are sometimes seen in finely executed work, wrought into 
the corner piece or into the title. In fact, the value of a 
map, as a means of conveying information of a topo¬ 
graphical nature, may often be greatly enhanced, espec¬ 
ially to the non-professional eye, by fine pen-and ink 


MAP DRAWING AND LETTERING. 


333 


sketches in perspective of portions of scenery or objects 
of special interest. But these, to be admissible, must be 
executed with that spirit and decision which mark the 
eye and hand of an artist. Of course there are maps, and 
maps, and the foregoing remarks are intended to apply 
mostly to such as are of large dimensions and intended 
for general inspection. For those of a strictly profes¬ 
sional nature, it is safe to say that the less exclusively 
ornamental work upon them, the more satisfactory will 
their appearance be. The dignity of an almost severe 
simplicity should pertain to a drawing addressed only to 
the professional eye, depending for its pleasing effect 
upon the sharpness and clearness of its execution, the 
completeness and beauty of the lettering, and its evident 
fitness to the purpose for which it was made, namely, 
business. This character of work is perfectly illustrated 
in many of the maps issued from the government depart¬ 
ments at Washington, and particularly in the charts of 
the United States Coast Survey. 

Border .—Having taken this general glance at the sub¬ 
ject, let us enter upon the consideration of the design and 
execution of the more important features of a properly 
finished map, and first, let us examine the border. 
Strictly speaking, this is a purely decorative feature 
as its entire omission would not affect the utility of the 
drawing. Still it aids materially in producing a happy 
effect, by limiting the eye at once to the considera¬ 
tion of those parts having a distinct claim upon its 
notice. The border bears a certain relation to the en¬ 
closed drawing, and is a sort of tribute or compliment to 
that which it encloses, and should in general reflect the 
character of the drawing. Thus a line drawing should 
have a border of geometrical design, executed with the 
same material as the drawing itself, and of greater or 
less complexity according to the extent and elaborateness 
of the work enclosed. It may be rectangular, or ellip¬ 
tical, or composed of both right lines and curves, ac¬ 
cording to the nature of the space enclosed, and the 




334 


A MANUAL OF LAND SURVEYING. 


draughtsman should never hesitate to omit or break the 
border in order to accommodate some projecting point 
or small portion of the plat with which its continuance 
would interfere. Should the drawing contain much free¬ 
hand work, the border might with propriety be partly 
free-hand also, but free-hand decorations representing 
garlands, vines, tassels, etc., and all rambling, sprawling 
decoration is always weak and in wretched taste. In all 
map drawing, color should be rigidly excluded from the 
border, as it gives a cheap and offensive conspicuousness 
to a subordinate feature. The most appropriate border 
for a map of moderate dimensions, and the one generally 
adopted by the best practice, consists simply of two lines, 
one heavy and the other, or inner one, light, with a white 
space between them about as wide as the heavy black 
line. The rule commonly followed is to make the total 
width of the border about the one-hundredth part of the 
shortest side of the map, supposing it to be a rectangle. 

The Meridian. — The next feature of a rightly con¬ 
structed map, and one which is by no means simply 
decorative but of very great practical importance, is the 
meridian. A true meridian is a necessary adjunct of 
all rightly constructed maps, as it is directionally their 
common line of comparison, and without it no just 
notion of the situation of the territory represented by 
the map, or of the bearings of its lines, can be obtained 
It is, in fact, one of the co-ordinates to which reference 
is made for the solution of all problems of position on 
the drawing, and as such is entitled to consideration. 
This line should therefore be a somewhat conspicuous 
object, and the object of its existence demands that it 
should not be so obscured by ornament as to defeat its 
use as a sharp, clear line of reference for all north and 
south lines. Nevertheless, the draughtsman is warranted 
in giving to its construction more than a hasty or careless 
consideration. It is usual to ornament the northern end 
of this meridian with some neatly drawn and character¬ 
istic device, such as an arrow-head, & fleur-de-lis, the head 
of a mediaeval lance, etc. At its southern extremity is 


MAP DRAWING AND LETTERING. 


335 


sometimes placed the feather end of an arrow or a cres¬ 
cent. Near the middle of the line may be drawn an east 
and west line, or four or eight pointed star, or radiating 
lines marking convenient points of graduation of the 
circle. It is well, also, to draw the magnetic meridian at 
the time of the survey, through the middle point of the 
true meridian, and mark the declination. This magnetic 
meridian should be even less ornamental than the true 
one, and when both are used it is generally agreed to 
draw a complete arrow-head on the latter, while the mag¬ 
netic line is subordinated by giving it only half a head, 
drawn on the right or left-hand side, as the declination is 
east or west. 

The construction of a meridian affords considerable 
opportunity for the display of skill and taste in the 
draughtsman. It may easily be made an attractive, sim¬ 
ple and elegant feature, reflecting the intelligence and 
spirit of an accomplished workman; or, by its awkward 
design and slovenly execution, shake one’s confidence in 
the mental capacity of one upon whom we should have a 
right to rely. Perhaps it would not be inappropriate to 
say that the meridian line should be sufficiently long, on 
most maps, to serve conveniently the purpose of trans¬ 
ferring its direction to other parts of the drawing by 
means of a triangle and straight-edge. The arrow-head 
at the vertex should be a sharply pointed figure, entirely 
different from the obtuse, nondescript object which too 
often offends the eye in that position. And, to avoid all 
possibility of mistake, it is well to place the letter N 
some slight distance above or below the arrow point. 
When a star is used to give the various points of the 
compass, its radiating arms should be narrow and slender, 
with sharp points, avoiding all appearance or suggestion 
of dullness. In short, the entire figure should be con¬ 
structed in the spirit of lightness and radiation, in har¬ 
mony with its office, which is simply that of indicating 
direction. 

Scale .—Of course it is entirely unnecessary to state that 
no map is complete, or prepared to accomplish the end 
and aim of its existence, without a scale. The scale is 


336 


A MANUAL OF LAND SURVEYING. 


here mentioned merely for the purpose of directing atten- 
tionto its paramount importance as furnishing one of 
the co-ordinates of a map. What is said later, with ref- 
ence to the title, applies equally to the treatment of 
scales. 

Lettering .—In no department of his work is the respon¬ 
sibility of the draughtsman greater, both in the matter of 
correctness of detail and beauty of appearance, than in 
that of lettering. Nothing more readily and strongly 
recommends a drawing to a favorable consideration than 
an appropriate and handsome style of lettering; while if 
poor, weak, uncertain and badly executed, it is apt to at 
once arouse in the mind an uncomfortable feeling of dis¬ 
trust and aversion, which surely detracts from the artistic 
value which the finished work should possess. 

There is a certain undefinable pleasure, a mental grati¬ 
fication, which we experience in looking upon anything 
well done; and this is particularly the case with all letter¬ 
ing upon maps, charts and drawings, as it conveys to the 
mind not only the sense of the words written, but beyond 
this there is indicated something of the mental character 
and practical experience of him to whom the work is due 
If it be strong, free and graceful in style, and sharp and 
clean in execution, it at once commands our confidence 
and respect, and saves us continual annoyance and waste 
of time in the endeavor to decipher some blurred or illeg¬ 
ible letter or figure, the failure to do which may put one 
to great inconvenience or delay. Thus, more perhaps 
than anything else, the lettering reflects the distinguish¬ 
ing qualities of the maker, and should be of such a 
nature as to clearly indicate a man completely and easily 
up to his work. 

Our alphabet is composed of a certain number of dis¬ 
tinct characters, or letters, and we also have certain 
additional characters for expressing numbers. It is not 
proposed to trace the origin and development of these 
characters, their evolution or history; it is sufficient to 
say that, so far as we are concerned now, they are arbi¬ 
trary signs which we are bound to adopt and follow. 
This is true, however, only as to their essential elements. 


MAP DRAWING AND LETTERING. 337 

Preserving these in their integrity, we are at liberty to 
alter and modify form and proportion very largely, 
according to individual fancy. By “ essential elements ” 
are meant those lines which are necessary to the recog¬ 
nition of the letter. Thus, every capital A, whatever 
else it may have, must have the three right-line elements 
which characterize it in its most simple form. 

Classification of Letters. — If one examines a book of 
specimen letters, he may conclude, from the bewildering 
variety of alphabets, that they are absolutely dissimilar, 
and subject to no system or order of classification. A 
little closer inspection, however, will reveal the fact that 
each letter of these alphabets, which apparently differ so 
widely, is constructed upon the common characteristic 
elements of that letter, a sufficient acquaintance with 
which, and their methods of variation, will enable one 
readily to design with consistency and uniformity the 
letters of any style of alphabet, which gives a most 
desirable independence of the mere devotion to copy. 

Rornan Letters .—Probably the most difficult letter to 
draw accurately and properly is the Roman, from which 
as a model we derive most if not all the alphabets now in 
use. This being so, possibly it will be well to enter some¬ 
what minutely into the discussion of its construction. 
Let us not, even here, entirely abandon our independence 
of judgment, but bear in mind that this style of letter in 
its present beauty of development is simply the resultant 
of the taste and practice of centuries; and that, while it 
is generally accepted as a standard, authorities differ as 
to its exact proportions and minor details, which relieves 
it from the undesirable rigidity of precise rule. 

Let us then, examine the Roman capitals. We can sep¬ 
arate this alphabet into three groups. Let the first embrace 
all letters whose essential elements are either horizontal or 
vertical right lines, such as I, E and II, six letters in all. 
The second group contains all letters having oblique 
right-line elements, as A, W, X and Y—nine; and the 
third group will comprise all letters into which curved 
elements enter, such as P, R, 8, etc., eleven in number. 


28 


338 


A MANUAL OF LAND SURVEYING. 


Aii examination of quo or two members of each group 
will be sufficient for our purpose. 

Let us take I, for instance, and by assigning it the 
proportions given in a set of letters designed by Prof. 
Warren, we make its height equal unity, its width one- 
fourtli the height; the caps at top and bottom project 
beyond the body of the letter a distance equal to half the 
width of the column of the letter. The complete propor¬ 
tions would then be: Height, unity; width of column, 
4-16tlis; total width, including caps, 8 lGths; projection of 
caps, 2-16ths; thickness of cap, l-16tli—these proportions 
to be preserved in all vertical columns of all letters. The 
letter L consists of an I with the addition of the arm at 
the bottom; this arm is 7-lGths in height, and the total 
width of the letter is 14-16ths. From these proportions 
the first group can readily be drawn. 

In the second group we meet with inclined elements, 
both heavy and light. Let it be noticed that in all letters 
having slanting heavy elements, with the exception of Z, 
all heavy elements slant downward to the right , and that 
these elements are 4-lGths perpendicular width; and all 
light elements have the same width as the caps, namely, 
1-lGtli. By properly proportioning the total widths of the 
letters, those of this section can be easily constructed. 
Observe that N is never drawn with a cap at the lower 
right-hand corner. 

In the third group, the widest part of curved elements 
equals the thickness of the columns, and the narrowest 
part equals the thickness of a cap. 

The letter just described is rather too heavy in its 
appearance, and would be much improved by reducing 
the width of all heavy elements to one-sixth of the total 
height, even one-seventli being sometimes used. 

The following table, taken from the last edition of 
Lieut. Smith’s “Topographical Drawing,” will serve as a 
guide in proportioning capital letters: 

“Taking the extreme width of II, measured across the 
middle, or exclusive cf the caps, as a unit, the widths of 
the other letters, or of their characteristic parts, may be 
expressed approximately by the numbers in the tliiid 


MAP DRAWING AND LETTERING. 


339 


column of the table. In case of letters having oblique 
lines, these widths are to be taken at the intersections of 
the outer oblique lines with the upper or lower limit of 
each letter. The caps are in all cases excluded.” 


Letter. 

Measured at 

H = l. 

Letter. 

Measured at 

H = 1. 

A 

Bottom .. 

1 l-16th. 

N 


7-8ths. 

B 

( Top. 

15-tfitlis. 

() 

Middle. 

r- 

X 

1 

} Bottom .. 

1 

P 

Top. 

15-16ths. 

n 

( Top. 

1 

Q 


1 l-8th. 

V 

j Bottom .. 

1 l-16th. 


\ Top. 

15-16ths. 

D 

Middle... 

1 l-16th. 

it 

} Bottom*... 

15-16ths. 


( Top. 

15-16ths. 

s 

\ Top. 

15-16tlis. 

E 

•J Middle... 

5-8ths. 

j Bottom .... 

1 


( Bottom .. 

1 

T 

Top. 

1 l-16th. 


( Tod. 

15-16ths. 

u 


7-8ths. 

F 

| Middle... 

5-8tlis. 

y 

Top. 

15-16ths. 


1 Top*. 

1 

w 

Top . 

l y 2 . 

Gr 

\ Bottom .. 

1 l-16th. 

v 

( Top. 

15-16ths. 

J 

Bottom .. 

3-4ths. 

A 

\ Bottom .... 

1 3-32nds. 


^ Top. 

15-16ths. 

Y 

Top. 

15-16ths. 

K 

\ Bottom .. 

1 l-16tli. 

y 

( Top. 

15-16ths. 

L 

Bottom .. 

1 

/j 

} Bottom .... 

1 

M 


1 3-32nds. 





^Measurement is from vertical tangent to the curve. 


The horizontal bars of H, E and F are at the middle of 
each letter; those of B, P and II are very slightly above 
it; while the horizontal bar of A is from one-eighth to 
one-sixth the height of the letter .below the middle. It 
improves the appearance of the E, C, G- and S to make 
the lower half a trifle wider than the upper, as is indi¬ 
cated by the proportions given in the table, and the ends 
of the arms of these letters should come nearly together, 
the extremities of the S being nearly on a horizontal line. 

Of course, no practical draughtsman, engaged in his 
work, stops to pay attention to these refinements of 
proportion. They serve simply as guides to the inexpe¬ 
rienced, and should be employed only in the study of the 
letter on a large scale. Letters of the size ordinarily em¬ 
ployed should be proportioned by the eye, and sketched 
in free-hand in pencil and afterwards inked in, instru- 
mentally if preferred. In drawing the small Roman 







































340 


A MANUAL OF LAND SURVEYING. 


letters, the same free-hand practice is required, and they 
are finally put in by a bold pressure of the pen for the 
heavy parts, the height of the small letters, such as a, e 
and o, being equal to three-fifths of the height of capitals. 

The general rule for spacing letters, is to make the 
areas between them approximately equal, and of course 
these areas are only estimated by the eye. 

Other Letters . — Having conquered the Roman, other 
alphabets will offer but few difficulties to the draughts¬ 
man. In the order of their importance for mapping 
purposes the letters are ranked thus: First, the upright 
Roman CAPITAL; second, the inclined CAPITAL; 
third, Roman, or ordinary small type; fourth, the small 
italic , or stump print , to be followed by the block, the 
skeleton, and an infinite variety of styles, fanciful and 
otherwise, not forgetting Old English. Inclined letters 
are usually regulated by a slope of three horizontal to 
eight vertical, and in the stub print the height of the 
smaller letters is three-fifths the height of capitals. This 
stub print somewhat resembles careful writing with the 
pen, and with some practice can be done very rapidly 
and made to present a very neat appearance. It is most 
perfectly adapted for statements, explanatory notes, etc. ; 
Free-hand random letters, of vigorous and graceful style, j 
may be easily designed, and sometimes relieve a drawing 
of undue stiffness. Good models in great variety can be ] 
easily obtained, and can with profit be carefully studied ' 
and practiced. 

Height .—The adjustment of the height of letters to the 
scale of the map is worth considering, and the following 
proportions have been suggested by Prof. McMillan, C. E.: 


SCALE. 

Height of Largest 
Upright Capitals. 

Height of Small Letters 
for Explanatory Notes, etc. 

1-GOOth, or 1 in. to 50 feet. 
l-2620th, or 2 ft. to 1 mile. 
l-5280th, or 1 ft. to 1 “ 

1-10560th, or 4 in. to 1 “ 

6-10ths inch. 
5-10ths “ 

4-10ths “ 

3-10ths “ 

12-t00ths inch. 
10-100ths “ 

8-100ths “ 

6-100ths “ 


The size and character of the letter used depends upon 
the nature and importance of the object to which it is 













MAP DRAWING AND LETTERING. 341 

applied, as capitals to large cities, important bodies of 
water, etc. On the United States Coast Survey charts 
upright letters are used for land objects, such as islands, 
points, etc., and for bodies of water, rivers, bays, and so 
on, the inclined letter is employed. 

When practicable the lines of lettering should be paral¬ 
lel to the base of the drawing ; if the lines incline toward 
the upper right-hand corner, the letters should be arranged 
to read from the bottom upward ; if they incline from 
the bottom upward to the left, they should read from the 
top downward. 

Title .—The last feature to which I shall direct atten¬ 
tion is the title, the execution of which alfords a suit¬ 
able opportunity for enhancing the beauty of the map 
by a choice selection of the letters used, an appropriate 
arrangement of the words, and the indulgence in a fit¬ 
ting amount of ornamentation. 

If the title be brief, and takes but a single line, it may 
be placed outside of and just below the border, otherwise 
it must be placed within it, its location depending upon 
the configuration of the map, preferably one of the cor¬ 
ners. The letters of the wording are varied in size 
according to the importance of the words. Their order 
of prominence is usually determined by answers to the 
following questions : First—What does the map repre¬ 
sent? Second—Where is the locality? Third—For 
whom have the survey and map been made? Fourth— 
By whom and when ? 

The title should be symmetrically arranged, with refer¬ 
ence to a middle vertical line, placing the most important 
words, if possible, about midway between the top and 
bottom of this arrangement; and the height of the let¬ 
ters composing this principal line should not exceed one 
thirty-third part of the shortest side of the map. 

In the case of an elaborate title it is a good plan to 
make a preliminary drawing, in which alterations can be 
made in the style and arrangement until the proper effect 
is produced. 

The title is the introduction of the drawing to the 
observer, and should be marked by a somewhat formal 


342 


A MANUAL OF LAND SURVEYING. 


and dignified grace. Anything of the nature of over¬ 
decoration should be carefully avoided, and I cannot 
close without expressing cordial antagonism toward the 
too frequent practice of employing elaborate pen-flour¬ 
ishes, “ rustic letters,” fluttering ribbons, and numerous 
devices of a like nature, for map decorations, with the 
laudable intention, no doubt, of adding enrichment to 
the drawing, but with the unfortunate results of cheap¬ 
ening the effect and violating good taste.” 




LEVELING. 


343 


CHAPTER XIV. 

LEVELING. 

1. Leveling is the operation of measuring the differ¬ 
ence in height of two or more points. 

The surface of water at rest is an example of a level 
surface. 

If the earth was a perfect sphere, a line of true level 
would be an arc or a circle having its centre at the centre 
of gravity of the earth. So far as common leveling is 
concerned it may be so considered, as the error arising 
therefrom is so small as to be of no practical consequence. 

The line of apparent level is a straight horizontal 
line passing through the point of observation, tangent to 
the line of true level. 

In precise leveling the difference between true and 
apparent level is measured, the instruments used are of 
the best, and all the operations are performed so as to re¬ 
duce the error to the smallest possible amount. In 
common leveling for streets, railroads, drains, water 
powers and the like operations, a lower degree of accuracy 
is required and the refinements of precise leveling are 
dispensed with. No attention is paid to The difference 
between true and apparent level, it being too small to 
affect the practical result. 

2. The deviation of the true from the apparent level 
between two points is equal to the square of the distance 
between the points, divided by the diameter of the earth. 

Also, The deviations for different distances are pro¬ 
portional to the squares of the distances. 


344 A MANUAL OF LAND SURVEYING. 

Calling the diameter of the earth 7920 miles and taking 
points one mile apart, we find the deviation = 0.000126 
miles = 0.665 ft. = 7.98 inches. For m miles, deviation = 
7.98 m 2 inches. 

The effect of the refraction of light is to apparently in¬ 
crease the difference between true and apparent level. 

For considerable distances the correction for curvature 
as above found is sometimes diminished by about one- 
sixth of itself. 


If the instrument is placed midway between the points 
whose difference in height is required, the errors are 
balanced and eliminate each other, giving a correct result. 


3. In leveling, two instruments are required, one to 
find a horizontal line, and the other to measure vertical 
distances. These instruments are called a Level and a 
Leveling rod. 

Level lines, for many common purposes, on a limited 
area, when no instruments are at hand, can be obtained 
by the following method : 


Suspend from some fixed point of support P by stout 
cords as indicated, a pole of any shape A B , having the 

longer end sharpened to a fine point. 
From this pole hang a heavy weight R 
as shown. Set two stakes S8 so that 
the point of the pole when swung 
around will just touch them. Smooth 
a place on each stake to receive marks 



Fig. 75. 


After taking the twist out of the supporting cord, care-’ 
fully swing the pole around and mark the exact place 
where the point of the pole touches each stake. Repeat 
this, and take the most satisfactory points. They will 
determine a level line of sight. 


A cheap instrument which almost anyone can make, 
having a more extended range, is made as follows : Take 






LEVELING. 


345 


' 7 b— ------ fl , two pieces of glass tubing three or four 

inches long and connect them with a 
rubber tube two or three feet long, so 
as to make a continuous water tight 
tube, with glass ends. Pass the ends 
of the tube through holes in a cross bar 
Fig. 76. made of a piece of board of suitable 

size, as shown in the cut, and fasten them with the tops 
projecting an inch or more above the bar. The cross bar 
may be fastened with a bolt and nut to a staff so that it 
may be set up and adjusted to a level line. Colored lluid 
is poured into the tube. The surface of the fluid in the 
glass tubes determines the level line. Sights of horse 
hair or fine wire may be attached close to the glass tubes 
and the cross bar adjusted to bring them into a level line. 

An instrument can thus be made at the expense of a 
few cents in money and a few minutes labor that will do 
very satisfactory work. 

4. If a tube be nearly filled with any liquid, as water, 
alcohol or ether, and closed, the liquid will seek the lowest 
part, arid the vacant space or bubble, as it is called, will be 
found at the highest part of the tube. If the tube is of 
glass, and very truly ground on the inside to a segment of 
a circle, it furnishes the best known means for determin¬ 
ing a level line. Such tubes are made and nearly filled 
with ether or alcohol, leaving a small space or bubble. 
When such a tube is placed convex side uppermost, the 
bubble seeks the highest point. Then a vertical line 
passing through the centre of the bubble will coincide 
with the radius of the arc to which the tube is ground. A 
perpendicular to this vertical line is a line of apparent 
level. Such a tube is the most essential part of the level. 
It is encased in a brass tube, having an opening so that 
the bubble and as much of the glass tube as necessary 
can be seen. A graduated scale is attached to it, or 
marked on the tube, by means of which the bubble is 
measured and its position with relation to other parts of 
the instrument is determined. The tube thus prepared is 
attached to a telescope in such a manner that it can be 








346 A MANUAL OF LAND SURVEYING. 

adjusted so as to bring the radius of the ground glass 
perpendicular to the line of sight in the telescope. 

The telescope is mounted in such a manner as to permit 
it to revolve freely in a horizontal plane and to be readily 
adjusted to the line of apparent level. 



Fig. 77. 


The plan 01 mounting the telescope most in favor in 
the United States is by a horizontal bar with forked 
arms called wyes. The telescope rests upon the wyes 
and is held in place by clips which may be loosened, per¬ 
mitting the telescope to be rolled over in the wyes. The 
bar is connected by a spindle to the tripod socket and 
leveling head similar to that used upon the transit. By 
permission of Messrs. Buff & Berger, of Boston, the 
following quotation is taken from their catalogue : 

5. “The Adjustments. —In a theoretically perfect 
level the following points are established : 

1. The object and eye-glasses are perpendicular to the 
optical axis at all distances apart. 

2. The optical axis coincides with the axis of rotation 
in the wyes. 

3. The axis of collimation coincides with the optical 
axis. 

4. The axis of collimation is parallel to the telescope 
level. 

5. The collars resting in the wyes are circles of the 
same diameter and concentric with the line of collima¬ 
tion of the telescope. 
































LEVELING. 


347 


6. The wyes are exactly similar, and similarly placed 
with reference to the line of collimation of the telescope. 

7. The level bubble moves over equal spaces for equal 
displacements of the telescope in altitude. 

8. The level bubble expands or contracts equally from 
the center in both directions, during changes of tempera¬ 
ture. 

9. The vertical axis of revolution is perpendicular to 
the line of collimation of the telescope. 

Of the above, the maker establishes points numbered 1, 
2, 5, G, 7 and 8. The remaining points, 3, 4 and 9, are 
established when the instrument leaves the shop, but 
being liable to derangement from rough usage, they are 
made adjustable in the held. 

Adjusting. After the engineer has set up the instru¬ 
ment and adjusted the eye-piece for parallax, the 
horizontal cross-line had better be made to lie in the 
plane of the azimuthal rotation of the instrument. This 
may be accomplished by rotating the reticule, after 
loosening the capstan-headed screws, until a point re 
mains bisected throughout the length of the line when 
the telescope is moved in azimuth. In making this ad¬ 
justment, the level tube is to be kept directly beneath the 
telescope tube. When made, the small set screw attached 
to one of the wyes may be set so that by simply bringing 
the projecting pin from the telescope against it, the cross- 
lines will be respectively parallel and perpendicular to 
the motion of the telescope in azimuth. 

The first collimating of the instrument may be made 
using an edge of some building, or any profile which is 
vertical. Make the vertical cross-line tangent to any 
such profile, and then turn the telescope half-way round 
in its wyes. If the vertical cross-line is still tangent to 
the edge selected, the vertical cross-line is collimated. 

Select some horizontal line, and cause the horizontal 
cross-line to be brought tangent to it. Again rotate the 
telescope half way round in its wyes, and if the horizontal 
cross-line is still tangent to the edge selected, the hori¬ 
zontal cross-line is collimated. 


348 


A MANUAL OF LAND SURVEYING. 


Having adjusted the two wires separately in this 
manner, select some well defined point which the cross- 
lines are made to bi-sect. Now rotate the telescope half 
way round in its wyes. If the point is still bi-sected, the 
telescope is collimated. A very excellent mark to use is 
the intersection of the cross-lines of a transit instrument. 

Center the eye-piece by the four capstan-headed screws 
nearest the eye end. This is done by moving the opposite 
screws in the same direction until a distant object under 
observation is without the appearance of a raise or fall 
throughout an entire rotation of the telescope in its 
wyes. The telescope is now adjusted. 

Next, bring the level bar over two of the leveling 
screws, focus the telescope upon some object about 300 
feet distant, and put on the sun-sliade. These precau¬ 
tions are necessary to a nice adjustment of the level tube. 
Throw open the two arms which hold the telescope down 
in its wyes, and carefully level the instrument over the 
two level screws parallel to the telescope. Lift the tele¬ 
scope out of its wyes, turn it end for end and carefully 
replace it. If the level tube is adjusted, the level will 
indicate the same reading as before. If it does not, cor¬ 
rect half the deviation by the two leveling screws and the 
remainder by moving the level tube vertically by means 
of the two cylinder nuts which secure the level tube to 
the telescope tube at its eye-piece end. Loosen the upper 
nut with an adjusting pin, and then raise or lower the 
lower nut as the case requires, and finally clamp that end 
of the level tube by bringing home the upper nut. This 
adjustment may require several repetitions before it is 
perfect. 

The level is now to be adjusted so that its axis may be 
parallel to the axis of the telescope. Rotate the telescope 
about 20° in its wyes, and note whether the level bubble 
has the same reading as when the bubble was under the 
telescope. If it has, this adjustment is made. If it has 
not the same reading, move the end of the level tube 
nearest the object-glass in a horizontal direction, when 
the telescope is in its proper position, by means of the 


LEVELING. 


349 


two small capstan headed screws which secure that end 
of the level to the telescope tube. If the level bubble goes 
to the object-glass end when that end is to the engineer’s 
right hand, upon rotating the telescope level toward him 
then these screws are to be turned in the direction of a 
left-handed screw, as the engineer sees them, and vice 
versa. Having completed this adjustment, the level bar 
itself must now be made parallel to the axis of the level. 

To do this, level the instrument carefully over two of 
its leveling screws, the other two being set as nearly leve 
as may be; turn the instrument 180° in azimuth, and if 
the level indicates the same inclination, the level bar is 
adjusted. If the level bubble indicates a change of 
inclination of the telescope in turning 180°', correct half 
the amount of the change by the two level screws, and 
the remainder by the two capstan-headed nuts at the end 
of the lever bar, which is to the engineer’s left hand when 
he can read the firm's name. Turn both nuts in the same 
direction, an equal part of a revolution, starting that nut 
first which is in the direction of the desired movement of 
the level bar. Many engineers consider this adjustment 
of little importance, preferring to bring the level bubble 
in the middle of its tube at each sight by means of the 
levelling screws alone, rather than to give any considera¬ 
tion to this adjustment, should it require to be made.” 

6. Leveling rods are made in a variety of styles and 
are of two principal classes, viz: target rods and speaking 
or self reading rods. 

Target rods are made of hard wood in two or more 
parts, which are grooved and tongued to slide upon each 
other, by which means they are lengthened out to 12 or 
more feet. They are graduated to feet, tenths and 
hundredths, the decimal notation being more convenient 
for computation than the division into inches and frac¬ 
tions of an inch. The target is a disc of brass made to 
slide up and down on the rod and to be clamped 
fast to the rod at any desired place. It is divided 
into quadrants painted alternately white and red. 
When used in leveling the target is moved up and down 


350 A MANUAL OF LAND SURVEYING. 

on the rod until the horizontal line between these divis¬ 
ions is brought to coincide with the line of sight in the 
level. The target has a vernier attached by which the 
distance on the rod is read to the nearest joVo part of a 
foot. In common leveling it is a useless refinement to 
carry the reading to thousandths of a foot, as it is out of 
harmony with the other conditions of the rod and the 
work to be done. The target on the rod, as a rule, is not 
capable of being set as closely and accurately to the level 
line as the vernier will read, nor will the rod be held so 
truly plumb as to justify so close a reading. Generally 
the line between the quadrants of the target is not per¬ 
pendicular to the rod and does not coincide with the 
zero of the vernier within several thousandths. 

Speaking rods are plain, 
straight rods, having the gradua¬ 
tions marked on them so boldly and 
distinctly that they can be read from 
the instrument. No targets are used 
with them, although some rods, like 
the Philadelphia rod, are made so as 
to be used either as target or speak¬ 
ing rods. There are many devices 
for marking the speaking rod, all of 
which are intended to facilitate ac¬ 
curate reading by the observer A 
simple form of graduation and letter¬ 
ing which gives excellent results in 
actual service is shown on a reduced 
scale in the cut. The graduations 
are to tenths and half tenths of a 
foot. Distances less than half a 
tenth are estimated by the eye. This 
is facilitated by having the figures 
for tenths made either .04 or .06 feet 
in length, and accurately spaced on 
the rod. 

The student having a level and a 
rod for use in practice may now 
solve the following problems in the 
field : 


Si 






Fig. 78. 























LEVELING. 


351 


7. Prob. 1. To find the difference of level of two points. 

Case 1 .— When the difference of level may be found by 
one setting of the instrument. 



Suppose A and B to be the points. Set up the level 
at a point about equidistant from A and B , though not 
necessarily in a line between them. Plant it firmly on 
the ground, placing the legs so as to bring the instrument 
nearly level, leaving as little as possible to be done with 
the leveling screws. If set up on yielding ground con¬ 
stant care will be required to be sure that the instrument 
is level at the instant the observation is taken. When 
the level is set up on ice or frozen ground, the legs will 
settle into the frost. It is well to set the instrument in 
the shade whenever convenient, as the rays of the sun, a 
passing cloud or a sudden breeze will throw the instru¬ 
ment out of level by causing unequal expansion and 
contraction of the metal. In precise leveling the instru¬ 
ment must be shaded. Having the instrument firmly 
planted, bring the telescope in line with one pair of the 
leveling screws and turn them in or out till the bubble is 
brought to the middle. Then bring the level in line with 
the other pair and again level it. Repeat until the bubble 
will remain in the middle of the tube through an entire 
revolution of the telescope around the spindle. 

The rod-man holds the rod at A, and its reading, Aa 
is taken. This is called a Back Sight. All observa¬ 
tions on other points taken at tile same setting of the 
instrument are termed Fore Sights. The distance Aa 
shows how much the line of collimation of the level is 
above the point A and is called the height of instru¬ 
ment. The rod-man now holds the rod on the point B 
and its reading is taken. The difference between the 
















352 A MANUAL OF LAND SURVEYING. 

back sight and the fore sight is the difference in height 
of the points A and B. If the back sight is 9.20 and the 
fore sight 6.40, then B is 2.80 higher than A. If the fore 
sight were 11.45 instead of 6.40, then B is 2.25 lower than 
A. The rod-man should stand square behind his rod and 
hold it plumb. Sometimes small levels are attached to 
the rod to plumb it by. If they are not used the leveler 
when necessary directs the rod-man to move the top of 
the rod to the right or left to plumb it that way, and the 
rod-man also moves it gently back and forth towards the 
level, until the smallest reading of the rod is obtained. 
It is manifest that as many points may be taken as can 
be reached from the instrument and that their relative 
heights will be shown by the distances they are below the 
horizontal plane of the instrument, which is told by the 
readings on the rod. 

Case 2 .—When the difference of level cannot be found 
by one setting of the instrument. 

Suppose A and E to be the points, and that it is 
necessary to set the instrument four times to find the 
difference between them. We find by the first setting 
the difference between the points A and B, as already 
described. We then go forward and find successively the 
differences between the points B and C, C and D, and 
T> and E. The algebraic sum of these differences is the 
difference in height of the points A and E. 

A convenient form of field notes in cases like the above 
consists of three columns as shown in the following 

Example. —Required the difference of level between 
the points A and E from the accompanying notes : 


Sta. 

Back Sights. 

Fore Sights. 

A 

3.2S 


B 

2.14 

7.15 

C 

3.25 

8.50 

D 

4.70 

3.45 

E 


2.75 


Which point is the higher and how much ? 









LEVELING. 


353 


A Bench Mark or Bench is a fixed point used for 
reference in finding- the heights of other points. It is 
indicated in the notes by the letters B. M. It is customary 
to establish bench marks at convenient distances along 
a line of levels by which the work may be reviewed, or at 
which it may be resumed after temporary cessation. The 
most convenient permanent objects are selected for the 
location of these bench marks, such as foundation stones 
in buildings, rocks or large boulders, or shoulders cut in 
the roots of large trees, so situated that the rod can be 
set up on them and the level readily taken. 

Where a line of levels is run taking a number of points 
it is customary to refer the heights to an assumed level 
plane called a datum. This is generally assumed to be 
far enough below the first or principal bench mark so 
that it shall be below the lowest station likely to be found 
in any part of the survey for which it is used. 

Negative heights are thus avoided: 

A line of levels is usually marked by stakes set at uni¬ 
form distances apart, marked and numbered consecutively 
from zero upwards. 100 feet is the distance most usually 
adopted between stations, although in levels for country 
drains it is sometimes found more convenient to space 
the stations by chains to correspond with the measures 
of the land surveys. Intermediate stakes are usually re¬ 
ferred to as plus stations , and are so marked on the stakes 
and in the notes. For instance, a stake set between 
stakes No. 6 and 7 at 40 feet from No. 0, is marked 6 + 40 
or simply +40. 

8. Prob. 2. To find the heights above a datum plane, 
of several stations on a given line. 

Suggestions.— Let AB (Fig. 80, page 354) be the given 
line and DP the datum plane assumed at any convenient 
distance, say 10 ft., below a bench near A. 

Set up the level at some convenient poir.t, for example 
between stations 2 and 3. 


X*4 



354 


A MANUAL OF LAND SURVEYING. 



Take the reading of the rod upon the bench and add it 
to the assumed height of the bench above the datum. 
The sum is the height of the instrument. 

Take the readings upon stations 0, 1, 2, 3, 4 and 5 in 
succession, and subtract each from the height of the 
instrument. The remainders are the heights, respect¬ 
ively, of those stations above the datum. 

Carry the instrument forward to another position, as 
between stations 6 and 7. 

Take the reading of the rod a second time on station 5, 
and add it to the height of station 5 as before found. 
The sum is the new height of instrument, with which 
proceed as before. 

A point used as station 5, as above indicated, is called 
a Turning Point. In practice, a bench is often adopted 
as a turning point. 

The reading of the rod upon a turning point or bench¬ 
mark is usually taken with somewhat greater precision 
than upon other points. 

A reading upon a bench or turning point is added to 
the height of the point above the datum in finding height 
of instrument; and a reading upon any point is sub¬ 
tracted from the height of instrument in finding the 
height of the point. 

Accordingly, an observation for the former is called a 
Plus Sight, denoted by -fS, and for the latter, a Minus 
Sight, denoted by —S. 


















LEVELING. 355 

The height of instrument is denoted by H. In., and the 
height of any point above the datum, by II. 

The following is an example of the notes made in solv¬ 
ing the above problem : 



9. Prob. 3. To find the cut or fill, to grade , at points 
between two given points. 

Suggestions. —Let A and B, (Fig. 80), denote the 
given points. Beginning at A, for example, measure the 
distance AB, at the same time marking it off into con¬ 
venient divisions of equal length, as 33 ft., 50 ft., 66 ft., or 
100 ft., for example, by driving pegs down to the surface 
of the ground. The last division will usually be fractional. 
Number the divisions, 0, 1, 2, 3, etc., beginning at A. 

Find now, (Prob. 2), the heights of the points, 0, 1, 2, 3, 
etc., above some convenient datum. 

For illustration, suppose the heights to be as given in 
the above Table (Prob. 2). Also suppose the height of the 
grade line at A to be 5 ft., and at B, 9 ft. 

The distance from A to B consisting of 8 equal parts, 
say of 50 ft., we should then have 

(9 ft. — 5 ft.) -r- 8 = 0.5 ft. = rise per station. 















356 


A MANUAL OF LAND SURVEYING. 


Beginning at A or station 0, we have 

7.98 — 5. = 2.98 = cut at 0 

9.13 — 5.50 = 0.63 = “ “ 1 

8.08 — 6.00= 2.08 = “ “ 2 

8.03 — 6.50= 1.53 = “ “ 3 

6.20 — 7.00 = —0.80 = fill “ 4 

9.67 — 7.50 = 2.17 = cut “ 5 

etc. etc. etc. 

Observe that we take the difference in height between 
the grade line and the station at each station; and since 
we have here proceeded frdm lower to higher points of 
the grade, we have added the rise of the grade per station 
to the height of the grade at the last preceding station. 

Let the student find the cut at each station, beginning 
at B, all other things being as above. 

Again, supposing the heights of the stations to be as 
above, let the student find the depths of cut and fill under 
the supposition that the height of the grade at A is 6 ft., 
and at B, 8.4 ft. 

lO. Drawing Profile. Big. 80 represents a section 
formed by a vertical plane passing through the points A 
and B , and meeting the datum plane in the line DP, 
The irregular line AB represents the intersection of the 
vertical plane with the surface of the ground, and 
is called the Profile. 

The manner of drawing the profile is as follows : 

Draw a horizontal line to represent the datum line, on 
which lay off to a convenient scale the distance between 
the stations. 

At the points of division of the datum line, erect per¬ 
pendiculars, on which lay off the surface heights of the 
several stations, in their order, but to a scale usually ten 
times greater than that used for the horizontal distances. 

A line drawn through the points thus located forms 
the profile. 

The use of a larger scale in drawing the vertical dis¬ 
tances serves to render the irregularities of the surface 


DRAINAGE SURVEYING. 357 

more apparent to the eye than they would be if drawn to 
the same scale with the horizontal measurements. 

The grade line is drawn through any two points at the 
proper distances from the datum line. The position and 
inclination of the grade line depend upon certain condi¬ 
tions required to be fulfilled by the work, such as the 
flowage of water, ease of travel, economy of construc¬ 
tion, etc. 

In road work the grade is often adopted with reference 
to an equalization of “cut” and “fill,” so that the mate¬ 
rial furnished by excavations shall make the embank¬ 
ments. The required position of the grade line, in order 
to fulfill this condition most advantageously, is conven¬ 
iently got by stretching a thread across the profile, varying 
the position of the thread until the areas intercepted by 
it and the profile on opposite sides appear to be equal. 

EXERCISES. 

11 . 1 . Find depths of cut or fill, and draw profile and 
grade line from the following notes : 


Sta. 

+ s. 

II. In. 

— S. 

II. 

II. Gr. 

Cut. 

Fill. 

0 

4.26 

14.26 


10.00 

8.00 



l 



6.30 





2 



8.45 





3 

4.12 


3.23 





340 


15.15 

8.20 





4 



4.63 



• 


5 



5.53 





5** 



5.75 


9.575 




Distance between stations, 100 It. 


2—5. Examples made by the student in the “ Field.” 

IT. DRAINAGE SURVEYING. 

12 . Of the many applications of leveling, the most 
common, perhaps, in the province of the ordinary sur¬ 
veyor, is that relating to drainage. Almost every neigh¬ 
borhood offers occasions for work of this kind. 















358 


A MANUAL OF LAND SURVEYING. 


13. Drains are of two forms: the Open Drain or 
Ditch, and the Under Drain. 

The former is adapted to the case of water lying upon 
he surface of the ground, and the latter to water under¬ 
lying the surface. Under drains are usually discharged 
into open drains, which are thus rendered an essential 
auxiliary to thorough drainage. 

14. Making the Survey. —This will he, in the first 
place, a careful reconnoissance of the locality respecting 
the general “lay of the land,” natural water courses, etc. 
In this will be determined the proper commencement, 
route and terminus of the drain. The term commence¬ 
ment will be here understood to mean the upper end of 
the drain, and terminus the outlet. The word commence¬ 
ment in connection with open drains will also be taken 
as significant of the proper place to begin the survey. 

Preliminaries having been settled, a stake marked 0 is 
driven at the point of commencement, and the survey, 
proper, begins by setting the transit over the stake and 
taking the bearings and distances of two convenient ob¬ 
jects near by as witnesses of the point of commencement. 
The location of the commencement should be described 
also by distances and direction from some neighboring 
monument or line of original survey. Thus, 10 ch. E. 
and 7.15 ch. N. of \ post bet. Secs. 11 and 14, T. 2 N.Tt. 5 E. 

These items are to be entered in the column of remarks 
in the Transit book, opposite the station 0. 

The instrument is then turned upon the first angle in 
the line of the drain and its bearing entered in the col¬ 
umn of bearings opposite station 0. 

Ax-men are required in clearing away bushes, making 
and driving stakes, etc. Two chain-men, the forward one 
carrying a transit-rod, now begin to measure at 0 in the 
direction of the first angle, and stakes marked 1, 2, 3, etc., 
are driven at uniform distances from each other. 

A 100-ft. tape is a convenient measure, and locates the 
stations at ordinarily suitable distances. 

A stake should be set also at each angle of the drain, 
and its distance from the last preceding station entered in 


DRAINAGE SURVEYING. 


359 


the notes. The points of meeting of any land-lines, roads, 
etc., should be noted by distances in a similar manner. 

The number of acres in farms whose lines are met 
may, very properly, be made a matter of memorandum 

The following is a specimen of the form of notes which 
are taken, in accordance with the above suggestions: 


TRANSIT NOTES. 




Distance 


Sta. 

Bearing. 

of 

Course. 

Remarks. 

0 

1 

2 

S. 70° E. 

U 

it 


0. A point 10 ch. E. and 7.15eh. N.of post 
on line bet. Secs. 11 and 14, T-, R- 

W. Oak 15, N. 23 1 / 2 ° E., 57 ft.; 

Hickory 12, S. 40° E., 34 ft. 

3 

4 

it 

it 


Land owned by John Doe, 80 A.; about 6 A. 
wet. 

5 

it 



528 

S. 2814° E. 

528 ft. 

5 28 . 1st Angle. 

6 

it 



7 

it 



8 

it 



840 

it 


8 4 <> Line bet. Secs. 13 and 24. 

9 

it 


B. Oak 10, S. 3514° W., 10 ft.; 

W. Ouk 18, N. 63° W., 28 ft. 

10 

11 

it 

it 


Richard Rowe, 160 A. on south, 30 A. 
swamp. 

1180 

East. 

652 ft. 

1 iso. 2d Angle. 

12 

it 



13 

it 



• 

it 



• 

it 



23 

it 



23« 

it 

1163 ft. 

23 4S . Terminus in drain by road side on 
Township line. 




Marked Boulder, N. 20° E., 15 ft., 

Ash 14, S. 27° W., 10 ft. 


4 



















360 


A MANUAL OF LAND SURVEYING. 


15. Taking the Levels. —The line of the drain hav¬ 
ing been established, the next thing is to take the levels. 
This is done in the manner previously described. Beside 
the engineer or principal surveyor, two men are required 
—a rod-man, and an ax-man to make and drive pegs. 

The pegs should be driven down even with the surface 
of the ground and at such a distance from the stakes 
marking the stations that they may be used without 
disturbance in excavating. Some practice driving them, 
say six inches, in front of the stakes; others set them 
opposite and at such a uniform distance from the record 
stakes as not to be disturbed by the digging. 

Bench marks should be made at convenient distances, 
for example at every tenth station, and far enough from 
the line not to be disturbed. 

16. Platting. —The field work having been completed, 
the next thing is to make a plat of the line and also of 
the sections or tracts of land which will be affected by 
the drain, writing the owner’s name and number of acres 
on each. On some convenient part of the plat, the 
courses and their corresponding distances should be 
noted, also the number of linear feet of drain on each 
separate tract. 

Next comes the drawing of the profile. This is most 
conveniently done by use of paper, called Profile paper, 
prepared specially for the purpose. Taking a piece of the 
proper width and of sufficient length to contain also the 
title and necessary explanatory notes, at the left hand, we 
begin on the edge next to us and write the numbers of all 
the stations in their order toward the right, upon the 
vertical lines. We then mark with the point of a sharp 
pencil the point of elevation of each station as taken 


361 


DRAINAGE SURVEYING. 

from the column of elevations in the level notes. Con¬ 
necting the points thus marked, by an ink line, we have 
the profile of the surface of the ground on the line of the 
drain. We then take a black thread and stretch it on 
the profile between the points assumed as grade , at the 
first and the last station. From this inspection, it will be 
seen whether it is necessary or desirable to introduce one 
or more changes of grade between the extreme points in 
order to avoid objectionable cuts. 

Having determined the situation of the grade lines, we 
then draw them in their places, preferably with red ink. 

Under the grade lines and upon the vertical lines of 
the several stations should be written in red ink the ele¬ 
vations of the grade, and below that, in black ink, the 
elevations of the surface. In a similar manner, above 
the profde may be written first, in red ink, the depths of 
the cuts, and, second, the widths of the ditch at bottom 
and top. 

The names of the land owners through whose land the 
ditch passes, with the number of linear feet on each, may 
be conveniently written upon the datum line. 

17 . The writer has saved himself and assistants a 
great many miles of tramping and wading through 
swamps and morasses in drainage surveys by running 
the transit and level lines for the drains both at one 
operation. It was found by repeated tests on long lines 
that the level on the transit gave very nearly if not quite 
as accurate results in leveling as the wye level. Hence 
the wye level was left at home and the transit line and 
levels were both run at the same time with the transit. 
A condensed form of keeping the notes was used, of 
which the following is a sample extract: 





362 A MANUAL OF LAND SURVEYING. 


Commencing at a point in the Section line 4.53 chains east of the 
quarter post between Sections it and 14, and running thence S. 16° W. 
Stations 2.00 chains apart. 


Sta. 

Obs. 

Ht. 

Inst. 

Elev. 

Grade 

Ht. 

Cut. 

Remarks. 

B. S. on 







B.M. 

4.96 

104.96 

100.00 



On Elm 40' to rt. of Sta. 1. 

0 

5.21 


99.75 

96.00 

3.75 

Elm and Black Ash. 

1 

5.30 


99.66 

95.90 

3.76 


2 

* 5.28 


99.68 

95.80 

3.88 

+50, enter thick Willows. 

3 

5.46 


99.50 

95.70 

3.80 


4 

5.72 


99.24 

95.60 

3.64 


5 

5.83 


99.13 

95.50 

3.63 


+60 

Angle rt. 12° 24' = S. 28° 24' W. 

Cross line fence between Smith 







and Jones. 

C 

5.84 


99.12 




B. S. 

2.91 

102.03 





G 

2.95 


99.08 

95.40 

3.68 

Open marsh. Saw grass. 

7 

3.06 


98.97 

95.30 

3.67 



All the rod readings are kept in one column. The back or plus 
sights, to he added to the elevation for height of instrument, are 
marked “ B. S.” The others are all to be subtracted from “ Ht. Inst.” 
for elevation of stations. 

IS. Depth and Width.—The depth of a drain obvi¬ 
ously depends upon the situation of the grade line with 
respect to the surface. In adjusting the grade line it is 
more important to guard against the drain being too 
shallow rather than too deep; most open drains are too 
shallow. 

Again, it should be taken into account, if the drain is 
to run through soft marshes and hard ridges, that the 
soft ground, on the withdrawal of the water, will settle; 
and so the drain may need to be dug deeper in some 
places than would otherwise be necessary. 

The necessary width of a drain of given depth and 
grade depends upon the quantity of water it is required 
to discharge in a given time. 


























DRAINAGE SURVEYING. 


363 


The width at the top is determined from the width at 
the bottom and the slope or inclination given the sides, 
which is usually from one to one and one-half feet on the 
horizontal to each foot n depth. 

19. Quantity of Discharge. —The amount of water 
which a drain may discharge in a given time obviously 
depends upon the area of the water-way or cross-section 
of the drain and the velocity of the stream. 

Thus, denoting by Q the quantity of discharge, by a 
the area of the water-way, and by v the mean velocity of 
discharge, we should have 


Q = av (1) 


As an approximate formula for computing the mean 
velocity of water Rowing in an open canal of uniform 
cross section and fall, Trautwine gives the formula 



in which V — mean velocity in feet per second, a = area 
of water-way in square feet,/= fall in feet per foot, and 
p — wet perimeter or the water border of the channel. 


Remark. —In applying the above formula, it is customary to use 
9000 for 8975 and .11 for .1089. 

Example .—Required the velocity and the capacity of a 
drain 5 ft. wide at the bottom, the sides having a slope of 


1 to 1, depth of water 3 ft., and the fall 2 ft. to 1000 ft. 

Solution .—Width at top == 5 ft. + 2 X 3 ft. = 11 ft. 
Area of water-way = H (11 ft. + 5 ft.) = 24 sq. ft. 
Wet perimeter = 5 ft. -f- 6/2 ft. = 13.5 ft. 

Fall per foot = 0.002 ft. 



Substituting in (2), V = 


Substituting in (1), Q = 24 X 5.55 = 133.2 cu. ft. per 
second, or 11,508,480 cu. ft. per day. 




364 


A MANUAL OF LAND SURVEYING. 


Trautwine gives also the following formula, with the 
remark that it is applicable also to sewers: 



a 


-X2-F| (3) 

V ) 


V 


in which a and p are as above described, and F is the fall 
in feet per mile. 

Remark.— In connection with the above formulas, as well as with 
others of similar import, Trautwine repeats again and again the 
caution that they are to be regarded only as approximately true. 

Table XII shows approximately the number of acres 
served by drains having bottom widths of 1 to 10 ft., with 
side slopes of 1 to 1, and various rates of fall per station, 
on the supposition of 1 inch rain-fall in 24 hours, one- 
half of which reaches the drain. 

20. Amount of Rainfall. — All calculations of 
requisite capacity of drains must be based upon the 
probable amount or number of inches of rainfall in a 
given time. The soil, however, acts as a reservoir up 
to the point of saturation, depending upon its texture, 
keeping from the drains altogether a portion of the 
rainfall, which passes off by evaporation or is absorbed 
l*y plants. 

The average annual rainfall in Michigan, Indiana, 
Illinois and Missouri is about 35 inches. In Ohio, for a 
period of ten years, it was reported to be 37.8G inches. 

In the matter of rainfall in Michigan, we are indebted 
to Prof. Carpenter for the following data: 

“ By a consultation of the meteorological records of the Agricul¬ 
tural College we learn that, although large showers in which the 
rainfall exceeds one inch occur comparatively seldom (on the average 
only four times a year), yet they bring with them twenty-eight per 
cent, of our total rainfall during that period, and consequently they 
must be fully provided for in any works for thorough drainage. The 
following table is compiled from the meteorological records kept at 
the college, and shows the comparative depth and number of showers 
from the months of March to December for five years. The last 
column shows the total percentage of rainfall in all the showers of a 
given depth. The last column but one shows the total percentage of 
the number of showers compared with the whole number. Although 


DRAINAGE SURVEYING. 


365 


this table is not extended sufficiently far back to give very accurate 
results, it is thought (since one year’s rainfall does not differ greatly 
from that of another year) to be sufficiently reliable to produce data 
for any ordinary case of farm drainage in this part of the United States 

TABLE OF SHOWERS FROM MARCH TO DECEMBER. 


Depth of Rain¬ 
fall in Inches. 

Number of Showers. 

Percentage of Total. 

1S72 

1873 

1874 

1875 

1876 

Total 

No. of 
Showers. 

Ain't of 
Rainfall. 

.00 to .25_ 

19 

40 

28 

35 

43 

165 

54.2 

17 

.25 to .50_ 

20 

14 

13 

9 

11 

67 

22.0 

21 

.50 to .75- - - 

G 

8 

6 

10 

5 

35 

11.5 

21 

.75 to 1.00_ 

2 

6 

5 

2 

3 

18 

06.0. 

13 

1.00 to 1.25 





2 

2 

00 7 

2 

1 95 to 1.50 

3 

2 



3 

8 

02 o 

9 

1 50 to 1 75_ 

1 


1 

1 


3 

01 0 

4 

1.75 to 2.00_ 


1 


1 


9 

00.7 

3 

2.00 to 2.25_ 








2.25 to 2.50_ 


1 




i 

00 3 

2 

2.50 to 2.75 

1 





i 

00 3 

2 

2.75 to 3.00 









3.00 to 3.25 




1 

1 

2 

00.7 

6 

Totals.- 






304 

100.00 

100 











“The amount of discharge of drains as compared with the rainfall is 
usually estimated at about 50 per cent. So that in order to produce 
thorough drainage it is necessary to assume that the capacity of the 
drains shall be sufficient to carry off during twenty-four hours one-half 
the water that fell the previous twenty-four hours. The probability of 
the rainfall in any day exceeding one inch is so slight that we shall be 
safe in assuming as the necessary carrying capacity of drains one-half 
of 3,630 cu. ft., or 1,815 cu. ft. of water for each acre drained.” 

21. Under Drains are formed in various ways; 
sometimes of brush, rails or loose stone trenched in, 
sometimes of tubes made of logs or of iron, sometimes 
of plank or of brick or stone laid in cement, and again of 
earthen tubes, of which there are various forms, called 
Tiles. 

The prevailing method of under-drainage for agricul¬ 
tural purposes consists in the use of cylindrical tiles, 
which are made of different sizes and usually about a 
foot in length. 

It is of this form of under drain, only, that we propose 
to write brietly. 




















































366 


A MANUAL OF LAND SURVEYING. 


22. Surveying for Under Drains. —Very much 
of what has been said upon surveying for the ditch or 
open drain applies also to the tile drain. The same pre¬ 
liminary inspection is required to determine the best 
location of the outlet and the proper directions of trunk 
and branch lines. Indications as to source of water, 
whether from springs on the premises or on lands sit¬ 
uated above, whether from rainfall, merely, upon the 
particular tract or also as flowing off from neighboring 
areas; the directions of slopes, whether of surface or of 
underlying strata; the character of the soil, etc., all 
have to be carefully observed and their bearing duly 
considered. 

23. Location of Drains.— As above intimated, any 
well Conducted survey for under drains contemplates the 
execution of a system of drains working together and 
depending upon each other. This will include usually a 
principal drain, called a Main, and lateral drains, called 
M inors, which discharge into the main. In an extended 
system, auxiliary mains called Sub-Mains are also 
introduced. 

Since it is the direct office of the minors to remove the 
surplus water from the ground, it is of the first import¬ 
ance that they be so located as successfully to perform 
their functions. To do this requires the exercise of care¬ 
ful judgment on the part of the engineer, respecting the 
proper directions of the minors and also their distances 
from each other. Equal care is requisite also in regard 
to the location of the main, so as properly to receive the 
water from the minors and discharge it at the principal 
outlet. 

As a rule, the main should be located at the foot of the 
regular slopes, or along the valleys of the field; and, in 
general, the minors should run directly down the slopes, 
discharging themselves obliquely into the main. 

Cases, however, will sometimes occur that require de¬ 
parture from the above rules, but these are to be regarded 
as “ exceptions which prove the rule.” 



DRAINAGE SURVEYING. 


367 


The distances of the minors from each other will be 
governed largely by the character of the soil as to per¬ 
meability, and to some extent by the depth of the drains. 
In a porous soil, as % general rule, the deeper the drain 
the further it will draw. 

Circumstances are infinitely varied. Every situation is 
a new one and must be treated on its own merits. None 
but the most general instruction on this point can be 
given in any treatise. About as practical a suggestion 
as may be afforded the student is, Go into the field and 
there mix plenty of brains loith your work. 

24. Running the Lines.— Having settled the ques¬ 
tion of the proper system of drains to be adopted, the 
next thing to be done is to lay out and measure the lines. 
This is perhaps most conveniently done in the case of 
under drains, by beginning at the outlet, measuring and 
staking out, first, the main lines of the system and then 
the branches. 

A distance of 50 ft. between stations is a convenient 
one in tile draining. In some instances, as where the fall 
is very slight, a less distance may be desirable; in others 
a greater one may give equally good results. In addition 
to the stakes driven at the uniform distances of the 
stations, a stake should mark the entrance of each minor, 
and the distance to it should be entered in the notes, 
in the usual manner. Such stakes mark the points of 
beginning in running out the minors. 

To facilitate examinations for “faults,” the points of 
entrance of the branches in the main drain should be 
established by witnesses. 

25. Taking the Levels.— This is done in the same 
manner as in the case of open drains, but, perhaps, with 
a somewhat greater degree of care and precision. The 
point assumed for the outlet must, of course, be suf¬ 
ficiently low to receive all the water of the field; and at 
the same time the outlet ought to be high enough to be 
at all times above the back water of the stream into 





368 


A MANUAL OF LAND SURVEYING. 


which the drain empties. A drain is of little more use 
under a violation of the latter condition than under a 
disregard of the former. 

In assuming the grade, due consideration must be had 
for proper depth consistently with required fall. 

The depth of an under drain should be, at the least, 
two feet; all the better if three or four feet in most soils. 

Henry F. French, author of ‘‘Farm Drainage,” says: 
“ We cannot, however, against the overwhelming weight 
of authority, and against the reasons for deeper drainage, 
which to us seem so satisfactory, conclude that even three 
feet is, in general, deep enough for under drains. Three- 
foot drains will produce striking results on almost any 
wet lands, but four-foot drains will be more secure and 
durable, will give wider feeding-ground to the roots, 
better filter percolating water, warm and dry the land 
earlier in Spring, furnish a larger reservoir for heavy 
rains, and, indeed, more effectually perform every office 
of drains.” 

Accordingly, the rule should be to approximate as 
closely as possible to what are thus regarded as desirable 
depths, admitting depths very much below the standard 
only when we must, in order to have any drains at all. 

Upon the question of necessary amount of fall, with 
which the surveyor is so often confronted in connection 
with the requirement of desirable depths, it is to be ob¬ 
served in the first place that large, deep streams require 
less fall than small ones; and, again, the form and the 
condition of the channel have much to do with the 
movement of water. 

“It has been found in practice that a water-course 
thirty feet wide and six feet deep will flow at the rate of 
one mile per hour, with a fall of no more than six inches 
per mile'' 

Examples are cited of successful operation of drains 
with three inches or even two and one-half inches fall to 
one hundred feet. 



DRAINAGE SURVEYING. 


369 


These, however, are to be regarded, probably, as excep¬ 
tional cases or as presenting, perhaps, the lowest limit 
that, even under the most favorable conditions of ordin¬ 
ary drainage, ought to be attempted. 

A very excellent authority says: “As to the fall neces¬ 
sary in tile draining, 1 consider one foot in one hundred 
yards the least fall to work upon with safety.” 

The above considerations will be perceived to bear 
upon the situation of the grade line, in order, on the one 
hand, to avoid too shallow drains, and, on the other, to 
secure the requisite fall for the proper movement of the 
water. 

Changes of grade, though undesirable, are admissible 
when not easily avoided. If possible, the heaviest grades 
should be in the direction of the outlet. When this 
cannot be, it may be desirable to introduce silt-wells at 
points of any considerable change of grade. 

The heights of the outlets of minor drains into the 
main are usually the heights of grade in the main drain 
for the same points. 

26. Constructing the Drain.— The principal 
point is the method of opening the trench and laying the 
tiles on the grade line. 

To do this systematically requires a measuring rod six 
or eight feet in length divided into feet, tenths, and hun¬ 
dredths of feet, the larger divisions being numbered up¬ 
ward, as in the ordinary leveling rod. A cord or wire, 
also, is needed, which is to be stretched above the line of 
the drain and adjusted to a position parallel to the grade 
line. This is done by inverting the measuring rod on the 
grade peg and bringing the cord or wire to the division 
of the rod indicating the cut at that point. The cord is 
thus placed at the full length of the measuring rod from 
the grade line or intended bottom of the trench. 

The cord may be held each fifty or one hundred feet by 
two slats, each about seven feet long, and movable about 


25 






370 


A MANUAL OF LAND SURVEYING. 


a bolt passing through a little distance from the upper 
end. These are called Shears. The cord or wire is pre¬ 
vented from slipping by a couple of turns, and is tied to 
a stake eight or ten feet from the shears. 

Another device consists in the use of stakes or posts 
driven on opposite sides of the ditch, and connected with 
a cross-bar arranged so that either end may be raised or 
lowered to a level, and fastened to the posts by a clamp 
and thumb-serew. The cross-bars being adjusted to the 
proper height, as above described, the cord or wire is 
drawn tightly across them, directly over the center line 
of the drain. 

Again, single stakes or posts, driven on one side of the 
ditch, each having attached at right angles an arm which 
maybe raised or lowered, and secured in place by a clamp 
and screw, are sometimes employed. 

By such means as the above, the ditch is readily dug to 
just the proper depth, and the tile laid to grade with ex¬ 
ceeding accuracy and with great rapidity. The proper 
distance from the top of the tile to the cord may be indi¬ 
cated by an arm attached to the measuring rod. 


27. Size of Tile.—The size of tile required in a 
given case will depend upon the quantity of water to be 
removed and the fall available to remove it. Formulas 
are given in works upon hydraulics, to express the veloci¬ 
ty and discharge of water flowing in pipes, but the condi¬ 
tions are so different in case of tiles that such formulas, 
at best, give only the most roughly approximate results. 


Thus, for example, the 
formula: 


F = 48 


following, which is Poncelet’s 
D X H \ Vi 
L + 54 7) ) 


in which, V — approximate velocity in feet per second, 
7) == diameter of pipe in feet, 77=total head in feet, and 
L — total length of pipe in feet. 



DRAINAGE SURVEYING. 


371 


Having found the velocity, we have 
Discharge in cu. ft. — vel. X cross section of pipe. 

Tables XIT and XITI are used for the above purpose, 
the latter quite extensively by drainage engineers and has 
been found to give good results. 

As regards size of tile for main and sub-main drains a 
good authority says, “ that can be regulated only by the 
person in charge of the drainage at any particular place, 
after seeing the land opened up and the minor drains dis¬ 
charging, As a rule, a circular pipe of three inches inter¬ 
nal diameter will discharge the ordinary drainage of six 
statute acres, and give sufficient space for the circulation 
of the air.” 

This estimate is based upon an amount of annual rain¬ 
fall of from twenty-six to thirty inches, which differs but 
slightly from that of Michigan and adjoining states. 

In addition to the above, it may be remarked that if the 
fall in the main is slight, a larger size of tile would be re¬ 
quired than if the fall was considerable, 

And, again in order to provide suitably for the accu¬ 
mulation of water which occurs toward the outlet, a 
larger size may be there required than that used in the 
upper part of the main. 

28. Protection at Outlets.— The outlets of under¬ 
drains should be protected by some construction to pre¬ 
vent the earth from falling down in front of the drain. 
A retaining wall of masonry laid in hydraulic cement is 
the best provision for the purpose. The outlets should 
be protected also by a coarse grating of some sort in 
front of the tile to prevent muskrats and other creatures 
from getting in. 

A common practice is to introduce at the outlet a box 
made of plank a few feet in length, into which the tile is 
made to discharge. 







372 


A MANUAL OF LAND SURVEYING. 


29. Silt Well. —This is a well sunk below thalevel 
of the tile for catching the silt gathered by the drains 
above it. It serves also the purpose of affording a means 
of inspecting the working of the drains. Silt wells may 
be constructed with a view, cliieliy, to facilitating the 
movement of the water at an abrupt bend in the drain. 
And again, they may be constructed somewhat with ref¬ 
erence to convenience of obtaining a pail of water for 
any purpose, in the field. 


ALPHABETS. 


373 


ABCDEF G H IJKL 
MNOPQRSTUVW 
X Y Z 

abcdefghijklmn 

opqrstuvwxyz 

i 234567890.,- 


A B CDEFGHIJK 
L M-N O P Q R S T U V 
WX YZ 

abcdefghijkl m n 
opqrstuvwxyz 

12 3 4 5 6 7 8 9 0 .,- 







374 


A MANUAL OF LAND SURVEYING. 


A B C E E F G- HIJKLM 
N □ P QRST U Y W X Y Z 

abed. Bfghi j klmnopq 

rstuvwxyz 

12345B7B90,, 

m m x i z 

a b c Ci e f n h iik I m n 
o jj q r s t u u m x tj z 
t254SG78a0. t F 




















SUGGESTIONS ON USE OF TABLES. 


I 


TABLES. 


SUGGESTIONS TO YOUNG SURVEYORS ON THE USES OF 

THE TABLES. 

Traverse Table. — The table calculated to quarter 
degrees is adapted to the simplest work of compass 
surveying, where great accuracy is neither required 
nor expected. When the transit is used, and the angles 
are taken to minutes or less, the author prefers the tables 
of logarithms and logarithmic sines and cosines to any 
traverse table yet made. They are capable of any re¬ 
quired degree of accuracy, and require the use of no more 
figures than the ordinary traverse table. In transit work, 
where latitudes and departures are to be calculated, it is 
well to refer the angles of all lines to a common base, 
just as in compass surveying all lines are referred to the 
meridian as a l^ase. Then, in any course, 

Latitude = co-sine of angle X length of the course. 

Departure = sine of angle X length of the course. 
Using the logarithmic tables, this is a short and simple 
computation. 

Example 1. —Angle, 36° 22'. Distance, 47.03. Required 
the latitude and departure. 

Log. of 47.63 = 1.077881 to which add 
log. sine, 30° 22' = 9.773018 

11.450899 the log of 28.24-f — departure. 

Log. of 47.03 = 1.077881 to which add 
log. cos., 30° 22' = 9.905925 

11.583800 the log. of 38.35-f = latitude. 

2. Course N. 57° 21' 20" E. 34.30X chains. Required 
the latitude and departure. 











II 


A MANUAL OF LAND SURVEYING. 


1. The Table of Tangents is convenient in 
estimating courses of lines to be run. 

Example 1 .—From the quarter post on the east side of 
Section 2 I wish to run a line for a road straight to a 
point 80 rods north of the southwest corner of Section 30. 
What course shall I run ? 

Solution .—Distance west, 5 miles; distance south, 4.25 
miles, which divided by 5 equals the natural tangent of 
the angle which the course makes with an east and west 
line, = .850. Find this number in the table of natural 
tangents and take out the corresponding angle, = 40° 22', 
which is the same as S. 49° 38 r W. 

2. What is the course from the village of Climax, at 
the east quarter post of Section 3, Township 3 south, 
Range 9 west, to the village of Richland, at the southwest 
corner of Section 14, Township 1 south, Range 10 west ? 
To the village of Schoolcraft, at the southeast corner 
of Section 19, T. 4 S., R. 11 W., from Climax? What to 
Schoolcraft from Richland? 

2. The Table of Secants is convenient for finding 
the hypothenuse of a triangle, thus simplifying many 
computations in the field. Secants not given in the table 
may be found by interpolation or by the formula: 

1 

Secant =-. 

cosine 

The following example indicates one of the practical 
applications in the field: 

Example .—Lots in a 
city are laid out with 
their lines perpendicu¬ 
lar to N Street and 
running through to M 
Street. Required the 
width (x) of the lots on 
M Street. 

Call the width of the 
lots on N Street r. 
Measure the angle A. 



Fig. 8i. 













SUGGESTIONS ON USE OF TABLES. Ill 

Then x = r, sec. A. If r — 100, as is common, x may be 
taken directly from the table. If r — 100, A = 21° 40', 
then x — 107.6. In laying out such lots it is generally 
easier and quicker to measure this distance on the street 
line than it is to set up the transit for each lot line and 
run it in. 

3. Table of Departures. — This table has many 
convenient uses, of which a few examples are given. 

Examples .—1. I wish to stake out a line along an old 
hedge row from quarter-post to section corner. On one 
side is a clear field. I go to the section corner, and make 
an offset of 25 links and set up a flag. I then go to the 
quarter-post, and, making an equal offset, find that I 
cannot see the flag; so I offset until I can see it—say 37 
links more. I sight to the flag, find from the table of 
departures the angle corresponding to 37 links at a dis¬ 
tance of 40 chains = 32', turn off the angle on the transit, 
and run the’ line back parallel with the section line, 
setting stakes on the true line, by 62 link offsets, as 
often as required. 

2. To run a true half-quarter-line when one end is 
inaccessible. 

Fig. 82 repre¬ 
sents t li e whole 
section, and ab the 
line to be run. 

Bisect % 7, setting 
stake at a. Meas¬ 
ure the angle acd , 
which we will call 
89° 24 r . By the 
field notes the 
north line of the 
section measures 
80.22, hence ac = 
20.05|. The south 
line measures 



Fig. 82. 
















IV 


A MANUAL OF LAND SURVF.YING. 


79.63, one-fourth of which is 19.90f. Hence the section 
line and half-quarter-line converge at the rate of 20.055 — 
19.9075 = .1475 chains per mile. From the table of 
departures we find the corresponding angle to be a little 
more than 6'. Hence we make the angle gab 6' greater 
than acd = 89° 30-K, and run the line accordingly. 

The foregoing are given as samples of many labor- 
saving uses of 'the tables, which the young surveyor 
should study out and be prompt to avail himself of 
when the occasion requires. 

TRIGONOMETRIC FUNCTIONS AND FORMULAE. 



Let Fig. 83 represent 
the various trigonomet¬ 
ric functions. Let ABC 
represent tjie angles, and 
abc the sides opposite in 
the right triangle formed 
by the radius, sine and 
cosine. Other parts as 
shown in the figure. 


Then sin A 

= BC 

cos A 

= AC 

tan A 

= DF 

cot A 

= H(I 

sec A 

= AD 

cosec A 

= AG 

versin A 

= CF 

coversin A 

BK 

exsec A 

■=BD 

coexsec A 

= BG 

chord A 

— BF 

chord 2 A 

O 

CM 

II 


Tables of these functions are calculated with radius 
AH = 1. 


a 

Sin A = — = cos 7? cos A 


c 


b 

— — = sin B 

c 


a 

Tan A = — = cot B cot A 
b 


b 

— — = tan B 
a 









V 


FORMULAE. 


• ' C 

Sec A = — = cosec B 
b 

c—b 

Yers A = -=- covers B 

c 


c 

cosec A = — = sec B 
a 

a—a 

coversin A —-= vers B 


c 


c — b 

Exsec A =-= cocxsec B 

b 


c — a 

coexsec A — -— exsec B 

a 


{ c sin A = b tan A 


a =■< 

c cos B 

= b cot B 


Wip + h ) ^ — 


a 

b 


sin A 

cos A 

c = H 

a 

b 


cos B sin B 

(_ -j/a 2 + b 2 


b=\ 


g cos A = a cot A 
c sin B = a tan B 

V(o + a) (c — a ) 


C = 90° = A+ 


ab 

Area = —. 



c 

b =-. 

cot A — cot B 

Useful in measur¬ 
ing heights of 
objects or passing 
obstacles in line. 


Fig. 84. 


























A MANUAL OF LAND SURVEYING. 


VI 


SOLUTION OF OBLIQUE TRIANGLES. 

Let ABC represent the angles, and abe the opposite 
sides, of any oblique triangle. 


Given. 

Sought. 

C= 180° — (A + B). 6= a sin B. 

sin A 

A, B, a 

C, b, 


c 

a 

c — sin A + B, 

sin A 

A t a, b 

B, C, 

sin A , 

B= - b. C = 180 — (A -f B\. 

CL 


c 

a 

e . sin C. 

sin A 

C , a,b 

VA+B) 

+ B)= 90° — \C. 


tA-B) 

tan \(A — B)= tan l(A -f B). 


A 

A = HA + B) + UA-B). 


B 

B = \(A + B)-\{A-B). 


e 

o - ( a + b) COS HA + B) = 

cos \(A — B) 

, N sin l (A + B) 

(a — b) 

v 7 sin i (A — B) 


Area 

Area = K=\ ab sin C. 

a, b, c 

A 

Let s -— ^ (ft —J- b —j- c), 



then si n i A =, K s ( s c ) 

\ be 



cos l A I s ( s a \ 

\ be 



tan \ A =_ /<* ~ ft > (* - «) 

\ *(#-a) 

sin A =f/ s (s ~ u > (s ~' ,) (s_c >- 


Area 

. , 2 (.s 1 — 6) (s — c 

vers in A = —-—- 

be 


Area = /s (s — o) ( 5 — 6) (6* — e). 

i,B , C, a 

Area 

. a 2 sin B sin C 

Area =-—. 

2 sin A 






























TABLE I. LOGARITHMS OF NUMBERS 


1 


TAB EES. 


LOGARITHMS OF NUMBERS 

FKOM 

1 TO 10000. 


N. 

- Gog. 

N. 

Log. 

N. 

Log. 

N. 

Log. 

1 

0 000000 

26 

1 414973 

51 

1 707570 

76 

1 880814 

2 

0 301030 

27 

1 431364 

52 

1 716003 

77 

1 886491 

3 

0 477121 

28 

1 447158 

53 

1 724276 

78 

1 892095 

4 

0 602060 

29 

1 462398 

54 

1 732394 

79 

1 897627 

5 

0 698970 

30 

1 477121 

55 

1 740363 

80 

1 903090 

6 

0.778151 

31 

1 491362 

56 

1 748188 

81 

1 908485 

7 

0 845098 

32 

1 505150 

57 

1 755875 

82 

1 913814 

8 

0 903090 

33 

1 518514 

58 

1 763428 

83 

1 919078 

9 

0 954243 

34 

1 531479 

59 

1 770852 

84 

1 924279 

10 

1 000000 

35 

1 544068 

60 

1 778151 

85 

1 929419 

11 

1 041393 

36 

1 556303 

61 

1 785330 

86 

1 934498 

12 

1 079181 

37 

1 568202 

62 

1 792392 

87 

1 939519 

13 

1 113943 

38 

1 579784 

63 

1 799341 

88 

1 944483 

14 

1 146128 

39 

l 591065 

64 

1 806180 

89 

1 919390 

15 

1 176091 

40 

1 602060 

65 

1 812913 

90 

1 954243 

16 

1 204120 

41 

1 612784 

66 

1 819544 

91 

1 959041 

17 

1 230449 

42 

1 623249 

67 

1 826075 

92 

1 963788 

18 

1 255273 

43 

1 633468 

68 

1 832509 

93 

1 968183 

19 

1 278754 

44 

1 643453 

69 

1 838849 

91 

1 973128 

20 

1 301030 

45 

1 653213 

70 

1 845098 

95 

1 977724 

21 

1 322219 

46 

1 662758 

71 

1 851258 

96 

1 982271 

22 

1 342423 

47 

l 672098 

72 

1 857332 

97 

1 986772 

23 

1 361728 

48 

1 681241 

73 

1 863323 

88 

1 991226 

24 

1 380211 

49 

1 690196 

74 

1 869232 

99 

1 995635 

25 

1 397940 

50 

1 698970 

75 

1 875061 

100 

2 000000 

















































2 TABLE I. LOGARITHMS OF NUMBERS. 


No. 

0 

1 

3 

3 

4 

5 

6 

7 

H 

9 

Diff. 

100 

000000 

000434 

000868 

001301 

001734 

002166 

002598 

003029 

003461 

003891 

432 

1 

4321 

4751 

5181 

5609 

6038 

6466 

6894 

7321 

7748 

8174 

428 

2 

8600 

9026 

9451 

9876 

010300 

010724 

011147 

011570 

011993 

012415 

424 

3 

012837 

013259 

013680 

014100 

4521 

4940 

5360 

5779 

6197 

6616 

'419 

4 

7033 

7451 

7868 

8284 

8700 

9116 

9532 

9947 

020361 

020775 

416 

5 

021189 

021603 

022016 

022428 

022841 

023252 

023664 

024075 

4486 

4896 

412 

C> 

5306 

5715 

6125 

6533 

6942 

7350 

7757 

8164 

8571 

8978 

408 

7 

9381 

9789 

030195 

030600 

031004 

031408 

031812 

032216 

032619 

033021 

404 

8 

033424 

033826 

4227 

4628 

5029 

5430 

5830 

6230 

6629 

7028 

400 

9 

742G 

7825 

8223 

8620 

9017 

9414 

9811 

040207 

040602 

040998 

396 

110 

041393 

041787 

042182 

042576 

042969 

043362 

043755 

044148 

044540 

044932 

393 

1 

5323 

5714 

6105 

6495 

6885 

7275 

7664 

8053 

8442 

8830 

389 

2 

9218 

9606 

9993 

050380 

050766 

051153 

051538 

051924 

052309 

052694 

386 

3 

053078 

053463 

053846 

4230 

4613 

4996 

5378 

5760 

6142 

6524 

382 

4 

6905 

7286 

7666 

8046 

8426 

8805 

9185 

9563 

9942 

060320 

379 

5 

0G0G98 

061075 

061452 

061829 

062206 

062582 

062958 

063333 

063709 

4083 

376 

G 

4458 

4832 

5206 

5580 

5953 

6326 

6699 

7071 

7443 

7815 

373 

7 

8186 

8557 

8928 

9298 

9668 

070038 

070407 

070776 

071145 

071514 

369 

8 

071882 

072250 

072617 

072985 

073352 

3718 

4085 

4451 

4816 

5182 

366 

9 

5547 

5912 

6276 

6640 

7004 

7368 

7731 

8094 

8457 

8819 

363 

120 

079181 

079543 

079904 

080266 

080626 

080987 

081347 

081707 

082067 

082426 

360 

1 

082785 

083144 

083503 

3861 

4219 

4576 

4934 

5291 

5647 

6004 

357 

2 

G3G0 

6716 

7071 

7426 

7781 

8136 

8490 

8845 

9198 

9552 

355 

3 

9905 

090258 

090611 

090963 

091315 

091667 

092018 

092370 

092721 

093071 

351 

4 

093422 

3772 

4122 

4471 

4820 

5169 

5518 

5866 

6215 

6562 

349 

5 

6910 

7257 

7604 

7951 

8298 

8644 

8990 

9335 

9681 

100026 

346 

G 

100371 

100715 

101059 

101403 

101747 

102091 

102434 

102777 

103119 

3462 

343 

7 

3804 

4146 

4487 

4828 

5169 

5510 

5851 

6191 

6531 

6871 

341 

8 

7210 

7549 

7888 

8227 

8565 

8903 

9241 

9579 

9916 

110253 

338 

9 

110590 

110926 

1112G3 

111599 

111934 

112270 

112605 

112940 

113275 

3609 

335 

130 

113943 

114277 

114611 

114944 

115278 

115611 

115943 

116276 

116608 

116940 

333 

1 

7271 

7603 

7934 

8265 

8595 

8926 

9256 

9586 

9915 

120245 

330 

2 

120574 

120903 

121231 

121560 

121888 

122216 

122544 

122871 

123198 

3525 

328 

3 

3852 

4178 

4504 

4830 

5156 

5481 

5806 

6131 

6456 

6781 

325 

4 

7105 

7429 

7753 

8076 

8399 

8722 

9045 

9368 

9690 

130012 

323 

5 

130334 

130655 

130977 

131298 

131619 

131939 

132260 

132580 

132900 

3219 

321 

G 

3539 

3858 

4177 

4496 

4814 

5133 

5451 

5769 

6086 

6403 

318 

7 

6721 

7037 

7354 

7671 

7987 

8303 

8618 

8934 

9249 

9564 

315 

8 

9879 

140194 

14050S 

140822 

141136 

141450 

141763 

142076 

142389 

142702 

314 

9 

143015 

3327 

3639 

3951 

4263 

4574 

4885 

5196 

5507 

5818 

311 

140 

146128 

146438 

146748 

147058 

147367 

147676 

147985 

148294 

148603 

148911 

S09 

1 

9219 

9527 

9835 

150142 

150449 

150756 

151063 

151370 

151676 

151982 

307 

2 

152288 

152594 

152900 

3205 

3510 

3815 

4120 

4424 

4728 

5032 

305 

3 

5336 

5640 

5943 

6246 

6549 

6852 

7154 

7457 

7759 

8061 

303 

4 

8362 

8664 

8965 

9266 

9567 

9868 

160168 

160469 

160769 

161068 

301 

5 

161368 

161667 

161967 

162266 

162564 

162863 

3161 

3460 

3758 

4055 

299 

6 

4353 

4650 

4947 

5244 

5541 

5838 

6134 

6430 

6726 

7022 

297 

7 

7317 

7613 

7908 

8203 

8497 

8792 

9086 

9380 

9674 

9968 

295 

8 

170262 

170555 

170848 

171141 

171434 

171726 

172019 

172311 

172603 

172895 

293 

9 

3186 

3478 

3769 

4060 

4351 

4641 

4932 

5222 

5512 

5802 

291 

150 

176091 

176381 

176670 

176959 

177248 

177536 

177825 

178113 

178401 

178689 

289 

1 

8977 

9264 

9552 

9839 

180126 

180413 

180699 

180986 

181272 

181558 

287 

2 

181844 

182129 

182415 

182700 

2985 

3270 

3555 

3839 

4123 

4407 

285 

3 

4691 

4975 

5259 

5542 

5825 

6108 

6391 

6674 

6956 

7239 

283 

4 

7521 

7803 

8084 

8366 

8647 

8928 

9209 

9490 

9771 

190051 

281 

5 

190332 

190612 

190892 

191171 

191451 

191730 

192010 

192289 

192567 

2846 

279 

6 

3125 

3403 

3681 

3959 

4237 

4514 

4792 

5069 

5346 

5623 

278 

i 

5900 

6176 

6453 

6729 

7005 

7281 

7556 

7832 

8107 

8382 

276 

8 

8657 

8932 

9206 

9481 

9755 

200029 

200303 

200577 

200850 

201124 

274 

9 

201397 

201670 

201943 

202216 

202488 

2761 

3033 

3305 

3577 

3848 

272 

No. 

O 

1 

•> 

3 

4 

5 

ti 

7 

8 

9 

Diff. 


















































































TABLE I. LOGARITHMS OF NUMBERS 


3 


No- 

1" - 

0 

1 

o 

3 

4 

5 

6 

7 

8 

9 

Diff. 1 

160 

204120 

204391 

204663 

204934 

205204 

205475 

205746 

206016 

206286 

206556 

271 

1 

6826 

7096 

7365 

7634 

7904 

8173 

8441 

8710 

8979 

9247 

269 

2 

9515 

9783 

210051 

210319 

210586 

210853 

211121 

211388 

211654 

211921 

267 

3 

212188 

212454 

2720 

2986 

3252 

3518 

3783 

4049 

4314 

4579 

266 

4 

4844 

5109 

5373 

5638 

5902 

6166 

6430 

6694 

6957 

7221 

264 

5 

7484 

7747 

8010 

8273 

8536 

8798 

9060 

9323 

9585 

9846 

262 

6 

220108 

220370 

220631 

220892 

221153 

221414 

221675 

221936 

222196 

222456 

261 

7 

2716 

2976 

3236 

3496 

3755 

4015 

4274 

4533 

4792 

5051 

259 

8 

5309 

5568 

5826 

6084 

6342 

6600 

6858 

7115 

7372 

7630 

258 

9 

7887 

8144 

8400 

8G57 

8913 

9170 

9426 

9682 

9938 

230193 

256 

170 

230449 

230704 

230960 

231215 

231470 

231724 

231979 

232234 

232488 

232742 

254 

1 

2996 

3250 

3504 

3757 

4011 

4264 

4517 

4770 

5023 

5276 

253 

2 

5528 

5781 

6033 

6285 

6537 

6789 

7011 

7292 

7544 

7795 

252 

3 

8046 

8297 

8548 

8799 

9049 

9299 

9550 

9800 

240050 

240300 

250 

4 

240549 

240799 

241048 

241297 

241546 

241795 

242044 

242293 

2541 

2790 

249 

5 

3038 

3286 

3534 

3782 

4030 

4277 

4525 

4772 

5019 

5266 

248 

6 

5513 

5759 

6006 

6252 

6499 

6745 

6991 

7237 

7482 

7728 

246 

7 

7973 

8219 

8464 

8709 

8954 

9198 

9443 

9687 

9932 

250176 

245 

8 

250420 

250664 

250908 

251151 

251395 

251638 

251881 

252125 

252368 

2610 

243 

9 

2853 

3096 

3338 

3580 

3822 

4064 

4306 

4548 

4790 

5031 

242 

180 

255273 

255514 

255755 

255996 

256237 

256477 

256718 

256958 

257198 

257439 

241 

1 

7679 

7918 

8158 

8398 

8637 

8877 

9116 

9355 

9594 

9833 

239 

2 

260071 

260310 

260548 

260787 

261025 

261263 

261501 

261739 

261976 

262214 

238 

3 

2451 

2688 

2925 

3162 

3399 

3636 

3873 

4109 

4346 

4582 

237 

4 

4818 

5054 

5290 

5525 

5761 

5996 

6232 

6467 

6702 

6937 

235 

5 

7172 

7406 

7641 

7875 

8110 

8344 

8578 

8812 

9046 

9279 

234 

6 

9513 

9746 

9980 

270213 

270446 

270679 

270912 

271144 

271377 

271609 

233 

7 

271842 

272074 

272306 

2538 

2770 

3001 

3233 

3464 

3696 

3927 

232 

8 

4158 

4389 

4620 

4850 

5081 

5311 

5542 

5772 

6002 

6232 

230 

9 

6462 

6692 

6921 

7151 

7380 

7609 

7838 

8067 

8296 

8525 

229 

190 

278754 

278982 

279211 

279439 

279667 

279895 

280123 

280351 

280578 

280806 

228 

1 

281033 

281261 

281488 

281715 

281942 

282169 

2396 

2622 

2849 

3075 

227 

2 

3301 

3527 

3753 

3979 

4205 

4431 

4656 

4882 

5107 

5332 

226 

3 

5557 

5782 

6007 

6232 

6456 

6681 

6905 

7130 

7354 

7578 

225 

4 

7802 

8026 

8249 

8473 

8696 

8920 

9143 

9366 

9589 

9812 

223 

5 

290035 

290257 

290480 

290702 

290925 

291147 

291369 

291591 

291813 

292034 

222 

6 

2256 

2478 

2699 

2920 

3141 

3363 

3584 

3804 

4025 

4246 

221 

7 

4466 

4687 

4907 

5127 

5347 

5567 

5787 

6007 

6226 

6446 

220 

8 

6665 

6884 

7104 

7323 

7542 

7761 

7979 

8198 

8416 

8635 

219 

9 

8853 

9071 

9289 

9507 

9725 

9943 

300161 

300378 

300595 

300813 

218 

2C0 

301030 

301247 

301464 

301681 

301898 

302114 

302331 

302547 

302764 

302980 

217 

1 

3196 

3412 

3628 

3844 

4059 

4275 

4491 

4706 

4921 

5136 

216 

2 

5351 

5566 

5781 

5996 

6211 

6425 

6639 

6854 

7068 

7282 

214 

3 

7496 

7710 

7924 

8137 

8351 

8564 

8778 

8991 

9204 

9417 

213 

4 

9630 

9843 

310056 

310268 

310481 

310693 

310906 

311118 

311330 

311542 

212 

5 

311754 

311966 

2177 

2389 

2600 

2812 

3023 

3234 

3445 

3656 

211 

6 

3867 

4078 

4289 

4499 

4710 

4920 

5130 

5340 

5551 

5760 

210 

7 

5970 

6180 

6390 

6599 

6809 

7018 

7227 

7436 

7646 

7854 

209 

8 

8063 

8272 

8481 

8689 

8898 

9106 

9314 

9522 

9730 

9938 

208 

9 

320146 

320354 

320562 

320769 

320977 

321184 

321391 

321598 

321805 

322012 

207 

210 

322219 

322426 

322633 

322839 

323046 

323252 

323458 

323665 

323871 

324077 

206 

1 

4282 

4488 

4694 

4899 

5105 

5310 

5516 

5721 

5926 

6131 

205 

2 

6336 

6541 

6745 

6950 

7155 

7359 

7563 

7767 

7972 

8176 

204 

3 

8380 

8583 

8787 

8991 

9194 

9398 

9601 

9805 

330008 

330211 

203 

4 

330414 

330617 

330819 

331022 

331225 

331427 

331630 

331832 

2034 

2236 

202 

5 

2438 

2640 

2842 

3044 

3246 

3447 

3649 

3850 

4051 

4253 

202 

6 

4454 

4655 

4856 

5057 

5257 

5458 

5658 

5859 

6059 

6260 

201 

7 

C4G0 

6660 

6860 

7060 

7260 

7459 

7659 

7858 

8058 

8257 

200 

8 

8456 

8656 

8855 

9054 

9253 

9451 

9650 

9849 

340047 

340246 

199 

9 

340444 

340642 

340841 

341039 

341237 

341435 

341632 

341830 

2028 

2225 

198 

No- 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Dili. I 




























































d. 


TABLE I. LOGARITHMS OP NUMBERS- 


No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diff. 

220 

342423 

342620 

342817 

343014 

343212 

343409 

343600 

343802 

343999 

344196 

197 

1 

4392 

4589 

4785 

4981 

5178 

5374 

5570 

5766 

5962 

6157 

196 

2 

6353 

6549 

6744 

6939 

7135 

7330 

7525 

7720 

7915 

8110 

195 

3 

8305 

8500 

8694 

8889 

9083 

9278 

9472 

9606 

9860 

350054 

194 

4 

350248 

350442 

350036 

350829 

351023 

351210 

351410 

351603 

351796 

1989 

193 

6 

2183 

2375 

2568 

2701 

2954 

3147 

3339 

3532 

3724 

3316 

193 

6 

4108 

4301 

4493 

4685 

4876 

5068 

5260 

5452 

5643 

5834 

192 

7 

G026 

6217 

6408 

6539 

6790 

6981 

7171 

7363 

7554 

7744 

191 

8 

7935 

8125 

8316 

8506 

8690 

8886 

9070 

9266 

9456 

9646 

190 

9 

9835 

360025 

360215 

300404 

360593 

360783 

360972 

361161 

361350 

361539 

189 

230 

301728 

361917 

362105 

362294 

362482 

362671 

362859 

363048 

363236 

363424 

188 

1 

3012 

3800 

3988 

4176 

4303 

4551 

4739 

4926 

5113 

6301 

188 

2 

5488 

5675 

5862 

6049 

6236 

6423 

6610 

6796 

6983 

7169 

187 

3 

7356 

7542 

7729 

791R 

8101 

8287 

8473 

8059 

8845 

9030 

186 

4 

9216 

9401 

9587 

9772 

9958 

370143 

370328 

370513 

370698 

370883 

185 

5 

371068 

371253 

371437 371622 

371806 

1991 

2175 

2360 

2544 

2728 

184 

6 

2912 

3096 

3280 

3464 

3647 

3831 

4015 

4198 

4382 

4565 

184 

7 

4748 

4932 

5115 

5298 

6481 

5664 

5846 

6029 

6212 

6394 

183 

8 

6577 

6759 

6942 

7124 

7306 

7488 

7670 

7852 

8034 

8216 

182 

9 

8398 

8580 

8761 

8943 

9124 

9306 

9487 

9668 

9849 

380030 

181 

240 

380211 

380392 

380573 

380754 

3C0934 

381115 

381296 

381476 

381656 

381837 

181 

1 

2017 

2197 

2377 

2557 

2737 

2917 

3097 

3277 

345b 

3636 

180 

2 

3815 

3995 

4174 

4353 

4533 

4712 

4891 

5070 

6249 

5428 

179 

3 

5606 

5785 

5904 

6142 

6321 

6499 

6677 

6856 

7034 

7212 

178 

4 

7390 

7568 

7746 

7923 

8101 

8279 

8456 

8634 

8811 

8989 

178 

5 

9166 

9343 

9520 

9698 

9875 

390051 

390228 

390405 

390582 

390759 

177 

6 

390935 

391112 

391288 

391464 

391641 

1817 

1993 

2109 

2345 

2521 

176 

7 

2697 

2873 

3048 

3224 

3400 

3575 

3751 

3926 

4101 

4277 

170 

8 

4452 

4027 

4802 

4977 

5152 

5326 

5501 

5676 

5850 

6025 

175 

9 

6199 

6374 

6548 

6722 

6896 

7071 

7245 

7419 

7592 

7766 

174 

250 

397940 

398114 

398287 

398461 

398634 

398808 

398981 

399154 

399328 

399501 

173 

1 

9674 

9847 

400020 

400192 

400365 

400538 

400711 

400883 

401056 

401228 

173 

2 

401401 

401573 

1745 

1917 

2089 

2261 

2433 

2605 

2777 

2949 

172 

3 

3121 

3292 

3404 

3635 

3807 

3978 

4149 

4320 

4492 

4663 

171 

4 

4834 

5005 

5176 

5346 

5517 

5088 

5858 

6029 

6199 

6370 

171 

5 

6540 

6710 

6881 

7051 

7221 

7391 

7561 

7731 

7901 

8070 

170 

6 

8240 

8410 

8579 

8749 

8918 

9087 

9257 

9426 

9595 

9764 

169 

7 

9933 

410102 

410271 

410440 

410609 

410777 

410946 

411114 

411283 

411451 

169 

8 

411020 

1788 

1956 

2124 

2293 

2401 

2629 

2796 

2964 

3132 

168 

9 

3300 

3467 

3635 

3803 

3970 

4137 

4305 

4472 

4639 

4806 

167 

200 

414973 

415140 

415307 

415474 

415641 

415808 

415974 

416141 

416308 

416474 

167 

1 

6041 

6807 

6973 

7139 

7306 

7472 

7638 

7804 

7970 

8135 

166 

i a 

8301 

8467 

8633 

8798 

8904 

9129 

9295 

9400 

9625 

9791 

165 

1 3 

9950 

420121 

420286 

420451 

420016 

420781 

420945 

421110 

421275 

421439 

165 

4 

421604 

1768 

1933 

2097 

2261 

2426 

2590 

2754 

2918 

3082 

164 

5 

3246 

3410 

3574 

3737 

3901 

4065 

4228 

4392 

4555 

4718 

164 

6 

4882 

6045 

5208 

5371 

5534 

6097 

5860 

6023 

6186 

6349 

163 

7 

6511 

6074 

6836 

6999 

7161 

7324 

7486 

7648 

7811 

7973 

162 

8 

8135 

8297 

8459 

8621 

8783 

8944 

9106 

9268 

9429 

9591 

162 

9 

9752 

9914 

430075 

430230 

430398 

430559 

430720 

430881 

431042 

431203 

161 

270 

431364 

431525 

431685 

431846 

432007 

432167 

432328 

432488 

432649 

32809 

161 

1 

2969 

3130 

3290 

3450 

3610 

3770 

3930 

4090 

4249 

4409 

160 

2 

4569 

4729 

4888 

5048 

5207 

6367 

5526 

5685 

5844 

6004 

159 

3 

6163 

6322 

6481 

6640 

6799 

6957 

7116 

7275 

7433 

7592 

159 

4 

7751 

7909 

8007 

8226 

8384 

8542 

8701 

8859 

9017 

9175 

158 

5 

9333 

9491 

9648 

9806 

9964 

440122 

440279 

440437 

440594 

440752 

158 

6 

440909 

441006 

441224 

441381 

441538 

1695 

1852 

2009 

2166 

2323 

157 

7 

2480 

2637 

2793 

2950 

3106 

3263 

3419 

3576 

3732 

3889 

157 

8 

4045 

4201 

4357 

4513 

4669 

4825 

4981 

5137 

5293 

5449 

156 

i ^ 

5004 

5700 

5915 

6071 

6226 

6382 

6537 

6692 

6848 

7003 

155 

Ino. 

*- -- 

© 

1 

2 

3 

4 

5 

6 

7 

8 

9 

m. 

















































TABLE I. LOGARITHMS OF NUMBERS, 


5 


No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diff. 

280 

447158 

447313 

447468 

447623 

447778 447933 

448088 

448242 

448397 

448552 

155 

1 

8706 

8861 

9015 

9170 

9324 

9478 

9633 

9787: 

9941 

450095 

154 

2 

450249 

450403 

450557 

450711 

450865 

451018 

451172 

451326 

451479 

1633 

154 

3 

1786 

1940 

2093 

2247 

2400 

2553 

2706 

2859 

3012 

3165 

153 

4 

3318 

3471 

3624 

3777 

3930 

4082 

4235 

4387 

4540 

4692 

153 

5 

4845 

4997 

5150 

5302 

5454 

5606 

5758 

5910 

6062 

6214 

152 

6 

6366 

6518 

6670 

6821 

6973 

7125 

7276 

7428 

7579 

7731 

152 

7 

7882 

8033 

8184 

8336 

8487 

8638 

8789 

8940 

9091 

9242 

151 

8 

9392 

9543 

9694 

9845 

9995 

460146 

460296 

460447 

460597 

460748 

151 

9 

460898 

461048 

461198 

461348 

461499 

1649 

1799 

1948 

2098 

2248 

150 

290 

462398 

462548 

462697 

462847 

462997 

463146 

463296 

463445 

463594 

463744 

150 

1 

3893 

4042 

4191 

4340 

4490 

4639 

4788 

4936 

5085 

5234 

149 

2 

5383 

5532 

5680 

5829 

5977 

6126 

6274 

6423 

6571 

6719 

149 

3 

6868 

7016 

7164 

7312 

7460 

7608 

7756 

7904 

8052 

8200 

148 

4 

8347 

8495 

8643 

8790 

8938 

9085 

9233 

9380 

9527 

9675 

148 

5 

9822 

9969 

470116 

470263 

470410 

470557 

470704 

470851 

470998 

471145 

147 

6 

471292 

471438 

1585 

1732 

1878 

2025 

2171 

2318 

2464 

2610 

147 

7 

2756 

2903 

3049 

3195 

3341 

3487 

3633 

3779 

3925 

4071 

146 

8 

4216 

4362 

4508 

4653 

4799 

4944 

5090 

5235 

5381 

5526 

146 

9 

5671 

5816 

5962 

6107 

6252 

6397 

6542 

6687 

6832 

6976 

145 

300 

477121 

477266 

477411 

477555 

477700 

477844 

477989 

478133 

478278 

478422 

145 

1 

8566 

8711 

8855 

8999 

9143 

9287 

9431 

9575 

9719 

9863 

144 

2 

480007 

480151 

480294 

480438 

480582 

480725 

480869 

481012 

481156 

481299 

144 

3 

1443 

1586 

1729 

1872 

2016 

2159 

2302 

2445 

2588 

2731 

143 

4 

2874 

3016 

3159 

3302 

3445 

3587 

3730 

3872 

4015 

4157 

143 

5 

4300 

4442 

4585 

4727 

4869 

5011 

5153 

5295 

5437 

557-1 

142 

6 

5721 

5863 

6005 

6147 

6289 

6430 

6572 

6714 

6855 

6997 

i42 

r' 

1 

7138 

7280 

7421 

7563 

7704 

7845 

7986 

8127 

8269 

8410 

341 

8 

8551 

8692 

8833 

8974 

9114 

9255 

9396 

9537 

9677 

9818 

141 

9 

9958 

490099 

490239 

490380 

490520 

490661 

490801 

490941 

491081 

491222 

140 

310 

491362 

491502 

491642 

491782 

491922 

492062 

492201 

492341 

492481 

492621 

140 

1 

2760 

2900 

3040 

3179 

3319 

3458 

3597 

3737 

3876 

4015 

139 

2 

4155 

4294 

4433 

4572 

4711 

4850 

4989 

5128 

5267 

5406 

139 

3 

5544 

5683 

5822 

5960 

6099 

6238 

6376 

6515 

6653 

6791 

139 

4 

6930 

7068 

7206 

7344 

7483 

7621 

7759 

7897 

8035 

8173 

138 

5 

8311 

8448 

8586 

8724 

8862 

8999 

9137 

9275 

9412 

9550 

138 

6 

9687 

9824 

9962 

500099 

500236 

500374 

500511 

500648 

500785 

500922 

137 

7 

501059 

501196 

501333 

1470 

1607 

1744 

1880 

2017 

2154 

2291 

137 

8 

2427 

2564 

2700 

2837 

2973 

3109 

3246 

3382 

3518 

3655 

136 

9 

3791 

3927 

4063 

4199 

4335 

4471 

4607 

4743 

4878 

5014 

136 

320 

505150 

505286 

505421 

505557 

505693 

505828 

505964 

506099 

506234 

506370 

136 

1 

6505 

6640 

6776 

6911 

7046 

7181 

7316 

7451 

7586 

7721 

135 

2 

7856 

7991 

8126 

8260 

8395 

8530 

8664 

8799 

8934 

9068 

135 

3 

9203 

9337 

9471 

9606 

9740 

9874 

510009 

510143 

510277 

510411 

134 

4 

510545 

510679 

510813 

510947 

511081 

511215 

1349 

1482 

1616 

1750 

134 

5 

1883 

2017 

2151 

2284 

2418 

2551 

2684 

2818 

2951 

3084 

133 

6 

3218 

3351 

3484 

3617 

3750 

3883 

4016 

4149 

4282 

4415 

133 

7 

4548 

4681 

4813 

4946 

5079 

5211 

5344 

5476 

5609 

5741 

133 

8 

5874 

6006 

6139 

6271 

6403 

6535 

6668 

6800 

6932 

7064 

132 

9 

7196 

7328 

7460 

7592 

7724 

7855 

7987 

8119 

8251 

8382 

132 

330 

518514 

518646 

518777 

518909 

519040 

519171 

519303 

519434 

519566 

519697 

131 

1 

9828 

9959 

520090 

520221 

520353 

520484 

520615 

520745 

520876 

521007 

131 

2 

521138 

521269 

1400 

1530 

1661 

1792 

1922 

2053 

2183 

2314 

131 

3 

2444 

2575 

2705 

2835 

2966 

3096 

3226 

3356 

3486 

3616 

130 

4 

3746 

3876 

4006 

4136 

4266 

4396 

4526 

4656 

4785 

4915 

130 

5 

5045 

5174 

5304 

5434 

5563 

5693 

5822 

5951 

6081 

6210 

129 

6 

6339 

6469 

6598 

6727 

6856 

6985 

7114 

7243 

7372 

7501 

129 

7 

7630 

7759 

7888 

8016 

8145 

8274 

8102 

8531 

8660 

8788 

129 

8 

8917 

9045 

9174 

9302 

9430 

9559 

9687 

9815 

9943 

530072 

128 

9 

530200 

530328 

530456 

530584 

530712 

530840 

530968 

531096 

531223 

1351 

128 

No. 

1 0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Dlff. 


?6 




















































6 TABLE I. LOGARITHMS OF NUMBERS. 


No 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diff. 

34C 

53147C 

531607 

531734 

531862 

531990 

532117 

532245 

532372 

53250C 

532627 

128 

1 

2754 

2882 

3009 

3136 

3264 

3391 

3518 

3645 

3772 

3899 

127 

2 

402C 

4153 

4280 

4407 

4534 

4661 

4787 

4914 

5041 

5167 

127 

3 

5294 

5421 

5547 

5674 

5800 

5927 

6053 

6180 

6306 

6432 

126 

4 

6558 

6685 

6811 

6937 

7063 

7189 

7315 

7441 

7567 

7693 

126 

5 

7819 

7945 

8071 

8197 

8322 

8448 

8574 

8699 

8825 

8951 

126 

6 

9076 

9202 

9327 

9452 

9578 

9703 

9829 

9954 

540079 

540204 

125 

7 

540329 

540455 

540580 

540705 

540830 

540955 

541080 

.541205 

1330 

1454 

125 

8 

1579 

1704 

1829 

1953 

2078 

2203 

2327 

2452 

2576 

2701 

125 

9 

2825 

2950 

3074 

3199 

3323 

3447 

3571 

3696 

3820 

3944 

124 

350 

544068 

544192 

544316 

544440 

544564 

544688 

544812 

544936 

545060 

545183 

124 

1 

5307 

5431 

5555 

5678 

5802 

5925 

6049 

6172 

6296 

6419 

124 

2 

6543 

6606 

6789 

6913 

7036 

7159 

7282 

7405 

7529 

7652 

123 

3 

7775 

7898 

8021 

8144 

8267 

8389 

8512 

8635 

8758 

8881 

123 

4 

9003 

9126 

9249 

9371 

9494 

9616 

9739 

9861 

9984 

550106 

123 

5 

550228 

550351 

550473 

550595 

550717 

550840 

550962 

551084 

551206 

1328 

122 

6 

1450 

1572 

1694 

1816 

1938 

2060 

2181 

2303 

2425 

2547 

122 

7 

2668 

2790 

2911 

3033 

3155 

3276 

3398 

3519 

3640 

3762 

121 

8 

3883 

4004 

4126 

4247 

4368 

4489 

4610 

4731 

4852 

4973 

121 

9 

5094 

5215 

5336 

5457 

5578 

5699 

5820 

5940 

6061 

6182 

121 

360 

556303 

556423 

556544 

556664 

556785 

556905 

557026 

557146 

557267 

557387 

120 

1 

7507 

7627 

7748 

7868 

7988 

8108 

8228 

8349 

8469 

8589 

120 

2 

8709 

8829 

8948 

9068 

9188 

9308 

9428 

9548 

9667 

9787 

120 

3 

9907 

500026 

560146 

560265 

560385 

560504 

560624 

560743 

560863 

560982 

120 

4 

561101 

1221 

1340 

1459 

1578 

1698 

1817 

1936 

2055 

2174 

119 

5 

2293 

2412 

2531 

2650 

2769 

2887 

3006 

3125 

3244 

3362 

119 

6 

3481 

3600 

3718 

3837 

3955 

4074 

4192 

4311 

4429 

4548 

119 

7 

4666 

4784 

4903 

5021 

5139 

5257 

5376 

5494 

5612 

5730 

118 

8 

5848 

5966 

6084 

6202 

6320 

6437 

6555 

6673 

6791 

6909 

118 

9 

7026 

7144 

7262 

7379 

7497 

7614 

7732 

7849 

7967 

8084 

118 

370 

568202 

568319 

568436 

568554 

568671 

568788 

568905 

569023 

569140 

569257 

117 

1 

9374 

9491 

9608 

9725 

9842 

9959 

570076,570193 

570309 

570426 

117 

2 

570543 

570660 

570776 

570893 

571010 

571126 

1243 

1359 

1476 

1592 

117 

3 

1709 

1825 

1942 

2058 

2174 

2291 

2407 

2523 

2639 

2755 

116 

4 

2872 

2988 

3104 

3220 

3336 

3452 

3568 

3684 

3800 

3915 

116 

5 

4031 

4147 

4263 

4379 

4494 

4610 

4726 

4841 

4957 

5072 

116 

6 

5188 

5303 

5419 

5534 

5650 

5765 

5880 

5996 

6111 

6226 

115 

7 

6341 

6457 

6572 

6687 

6802 

6917 

7032 

7147 

7262 

7377 

115 

8 

7492 

7607 

7722 

7836 

7951 

8066 

8181 

8295 

8410 

8525 

115 

9 

8639 

8754 

8868 

8983 

9097 

9212 

9326 

9441 

9555 

9669 

114 

380 

579784 

579898 

580012 

580126 

580241 

580355 

580469 

580583 

580697 

580811 

114 

1 

580925 

581039 

1153 

1267 

1381 

1495 

1608 

1722 

1836 

1950 

114 

2 

2063 

2177 

2291 

2404 

2518 

2631 

2745 

2858 

2972 

3085 

114 

3 

3199 

3312 

3426 

3539 

3652 

3765 

3879 

3992 

4105 

4218 

113 

4 

4331 

4444 

4557 

4670 

4783 

4896 

5009 

5122 

5235 

5348 

113 

5 

5461 

5574 

5686 

5799 

5912 

6024 

6137 

6250 

6362 

6475 

113 

6 

6587 

6700 

6812 

6925 

7037 

7149 

7262 

7374 

7486 

7599 

112 

7 

7711 

7823 

7935 

8047 

8160 

8272 

8384 

8496 

8608 

8720 

112 

8 

8832 

8944 

9056 

9167 

9279 

9391 

9503 

9615 j 

9726 

9838 

112 

9 

9950 

590061 

590173 

590284 

590396 

590507 

590619 

590730 

590842 

590953 

112 

390 

591065 

591176 

591287 

591399 

591510 

591621 

591732 

591843 

591955 

592066 

111 

1 

2177 

2288 

2399 

2510 

2621 

2732 

2843 

2954 

3064 

3175 

111 

2 

3286 

3397 

3508 

3618 

3729 

3840 

3950 

4061 

4171 

4282 

111 

3 

4393 

4503 

4614 

4724 

4834 

4945 

5055 

5165 

5276 

5386 

110 

4 

6496 

5606 

5717 

5827 

5937 

6047 

6157 

6267 

6377 

6487 

110 

5 

6597 

6707 

6817 

6927 

7037 

7146 

7256 

7366 

7476 

7586 

110 

6 

7695 

7805 

7914 

8024 

8134 

8243 

8353 

8462 

8572 

8681 

110 

7 

8791 

8900 

9009 

9119 

9228 

9337 

9446 

9556 

9665 

9774 

109 

8 

9883 

9992 

600101 

600210 

600319 

600428 

600537 

600646 

600755 

600864 

109 

9 

600973 

601082 

1191 

1299 

1408 

1517 

1625 

1734 

1843 

1951 

109 

No. 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diff. 





































































TABLE I. LOGARITHMS OP NUMBERSo 7 


No. 

0 

1 

o 

3 

4 

5 

6 

7 

8 

ft 

Diff. 

400 

602060 

602169 

602277 

602386 

602494 

602603 

602711 

602819 

602928 

603036 

108 

1 

3144 

3253 

3361 

3169 

3577 

3686 

3794 

3902 

4010 

4118 

108 

2 

4226 

4334 

4442 

4550 

4658 

4766 

4874 

4982 

5089 

5197 

108 

3 

5305 

5413 

5521 

5628 

5736 

5844 

5951 

6059 

6166 

6274 

108 

4 

6381 

6489 

6596 

6704 

6811 

6919 

7026 

7133 

7241 

7348 

107 

5 

7455 

7562 

7669 

7777 

7884 

7991 

8098 

8205 

8312 

8419 

107 

G 

8526 

8633 

8740 

8847 

8954 

9061 

9167 

9274 

9381 

9488 

107 

7 

9594 

9701 

9808 

9914 

610021 

610128 

610234 

610341 

610447 

610554 

107 

8 

610660 

610767 

610873 

610979 

1086 

1192 

1298 

1405 

1511 

1617 

106 

9 

1723 

1829 

1936 

2012 

2148 

2254 

2360 

2466 

2572 

2678 

106 

410 

612784 

612890 

612996 

613102 

613207 

613313 

61.3419 

613525 

613630 

613736 

106 

1 

3842 

3947 

4053 

4159 

4264 

4370 

4475 

4581 

4686 

4792 

106 

2 

4897 

5003 

5108 

5213 

5319 

5424 

5529 

5634 

5740 

5K45 

105 

3 

5950 

6055 

6160 

6265 

6370 

6476 

6581 

6686 

6790 

6895 

105 

4 

7000 

7105 

7210 

7315 

7420 

7525 

7629 

7734 

7839 

7943 

105 

5 

8048 

8153 

8257 

8362 

8466 

8571 

8676 

8780 

8884 

8989 

105 

G 

9093 

9198 

9302 

9106 

9511 

9615 

9719 

9824 

9928 

620032 

104 

7 

620136 

620240 

620344 

620448 

620552 

620656 

620760 

620864 

620968 

1072 

104 

8 

1176 

1280 

1384 

1488 

1592 

1695 

1799 

1903 

2007 

2110 

104 

9 

2214 

2318 

2421 

2525 

2628 

2732 

2835 

2939 

3042 

3146 

104 

420 

623249 

623353 

623156 

623559 

623663 

623766 

623869 

623973 

624076 

624179 

103 

1 

4282 

4385 

4488 

4591 

4695 

4798 

4901 

5004 

5107 

5210 

103 

2 

5312 

5415 

5518 

5621 

5724 

5827 

5929 

6032 

61&5 

6238 

103 

3 

6340 

6443 

6.546 

6618 

6751 

6853 

6956 

7058 

7161 

7263 

103 

4 

7366 

7468 

7571 

7673 

7775 

7878 

7980 

8082 

8185 

8287 

102 

5 

8389 

8491 

8593 

8695 

8797 

8900 

9002 

9104 

9206 

9308 

102 

6 

9410 

9512 

9613 

9715 

9817 

9919 

630021 

630123 

630224 

630326 

102 

7 630428 

630530 

630631 

630733 

630835 

630930 

1038 

1139 

1241 

1342 

102 

8 

1444 

1545 

1647 

1748 

1849 

1951 

2052 

2153 

2255 

2356 

101 

9 

2457 

2559 

2660 

2761 

2862 

2963 

3064 

3165 

3266 

3367 

101 

430 633468 

633569 

633670 

633771 

633872 

633973 

634074 

634175 

634276 

634376 

101 

1 

4477 

4578 

4619 

4779 

4880 

4981 

5081 

5182 

5283 

5383 

101 

2 

5484 

5584 

5685 

5785 

5886 

5986 

6087 

6187 

6287 

6388 

100 

3 

6488 

6588 

6688 

6789 

6889 

6989 

7089 

7189 

7290 

7390 

100 

4 

7490 

7590 

7690 

7790 

7890 

7990 

8090 

8190 

8290 

8389 

100 

5 

8489 

8589 

8689 

8789 

8888 

8988 

9088 

9188 

9287 

9387 

99 

6 

9486 

9586 

9686 

9785 

9885 

9984 

640084 

640183 

640283 

640382 

99 

7 640481 

640581 

610680 640779 

640879 

640978 

1077 

1177 

1276 

1375 

99 

8 

1474 

1573 

1672 

1771 

1871 

1970 

2069 

2168 

2267 

2366 

99 

9 

2465 

2563 

2662 

2761 

2860 

2959 

3058 

3156 

3255 

3354 

99 

440 

643453 

643551 643650 

643749 

643847 

643946 

644044 

644143 

644242 

644340 

98 

1 

4439 

4537 

4636 

4734 

4832 

4931 

5029 

5127 

5226 

5324 

98 

2 

5422 

5521 

5619 

5717 

5815 

5913 

6011 

6110 

6208 

6306 

98 

3 

6404 

6502 

6600 

6698 

6796 

6894 

6992 

7089 

7187 

7285 

98 

4 

7383 

7481 

7579 

7676 

7774 

7872 

7969 

8067 

8165 

8262 

98 

5 

8360 

8458 

8555 

8653 

8750 

8848 

8945 

9043 

9140 

9237 

97 

6 

9335 

9432 

9530 

9627 

9724 

9821 

9919 

650016 

650113 

650210 

97 

7 650308 

650405 

650502 

650599 

650096 

650793 

650890 

0987 

1084 

1181 

97 

8; 

1278 

1375 

1472 

1569 

1666 

1762 

1859 

1956 

2053 

2150 

97 

9 

2246 

2343 

2140 

2536 

2633 

2730 

2826 

2923 

3019 

3116 

97 

450 653213 

653309 

653105 

653502 

653598 

653695 

653791 

653888 

653984 

654080 

96 

1 

4177 

4273 

4369 

4465 

4562 

4658 

4754 

4850 

4946 

5042 

96 

2 

5138 

5235 

5331 

5427 

5523 

5619 

5715 

5810 

5906 

6002 

96 

3 

6098 

6194 

6290 

6386 

6482 

6577 

6673 

6769 

6864 

6960 

96 

4 

7056 

7152 

7247 

7343 

7438 

7534 

7629 

7725 

7820 

7916 

96 

5 

8011 

8107 

8202 

8298 

8393 

8488 

8584 

8679 

8774 

8870 

95 

6 

8965 

9060 

9155 

9250 

9346 

9441 

9536 

9631 

9726 

9821 

95 

7 

9916 

660011 

660106 

660201 

660296 

660391 

660486 

660581 

660676 

660771 

95 

8 

660865 

0960 

1055 

1150 

1245 

1339 

1434 

1529 

1623 

1718 

95 

9 

1813 

1997 

2002 

2096 

2191 

2286 

2380 

2475 

2569 

2663 

95 

No.! 

0 

1 

2 

3 

4 

5 

« 

7 

8 

ft 

Diff. 



































































































8 


TARLE I. LOGARITHMS OF NUMBERS, 


No. 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diff. 

460 

662758 

662852 

662947 

663041 

663135 

663230 

663324 

663418 

663512 

663607 

94 

1 

3701 

3795 

3889 

3983 

4078 

4172 

4266 

4360 

4454 

4548 

94 

2 

4642 

4736 

4830 

4924 

5018 

5112 

5206 

5299 

5393 

5487 

94 

3 

6581 

5675 

5769 

5862 

5956 

6050 

6143 

6237 

6331 

6424 

94 

4 

6518 

6612 

6705 

6799 

6892 

6986 

7079 

7173 

7266 

7360 

94 

5 

7453 

7546 

7640 

7733 

7826 

7920 

8013 

8106 

8199 

8293 

93 

6 

8386 

8479 

8572 

8665 

8759 

8852 

8945 

9038 

9131 

9224 

93 

7 

9317 

9410 

9503 

9596 

9689 

9782 

9875 

9967 

670060 

670153 

93 

8 

670246 

670339 

670431 

670524 

670617 

670710 

670802 

670895 

0988 

1080 

93 

9 

1173 

1265 

1358 

1451 

1543 

1636 

1728 

1821 

1913 

2005 

93 

470 

672098 

672190 

672283 

672375 

672407 

672560 

672652 

672744 

672836 

672929 

92 

1 

3021 

3113 

3205 

3297 

3390 

3*82 

3574 

3606 

3758 

3850 

92 

2 

3942 

4034 

4126 

4218 

4310 

4402 

4494 

4586 

4677 

4769 

92 

3 

4861 

4953 

5045 

5137 

5228 

5320 

5412 

5503 

5595 

5687 

92 

4 

5778 

5870 

5962 

6053 

6145 

6236 

-6328 

6419 

6511 

6602 

92 

5 

6694 

6785 

6876 

6968 

7059 

7151 

7242 

7333 

7424 

7516 

91 

6 

7607 

7698 

7789 

7881 

7972 

8063 

8154 

8245 

8336 

8427 

91 

7 

8518 

8609 

8700 

8791 

8882 

8973 

9064 

9155 

9246 

9337 

91 

8 

9428 

9519 

9610 

9700 

9791 

9882 

9973 

680063 

680154 

680245 

91 

9 

680336 

680426 

680517 

680607 

680698 

680789 

680879 

0970 

1060 

1151 

91 

480 

681241 

681332 

681422 

681513 

681603 

681693 

681784 

681874 

681904 

682055 

90 

1 

2145 

2235 

2326 

2416 

2506 

2596 

2686 

2777 

2867 

2957 

90 

2 

3047 

3137 

3227 

3317 

3407 

3497 

3587 

3677 

3767 

3857 

90 

3 

3947 

4037 

4127 

4217 

4307 

4396 

4486 

4576 

4666 

4756 

90 

4 

4845 

4935 

5025 

5114 

5204 

5294 

5383 

5473 

5563 

5652 

90 

5 

5742 

5831 

5921 

6010 

6100 

6189 

6279 

©368 

6458 

6547 

89 

6 

6636 

6726 

6815 

6904 

6994 

7083 

7172 

'1251 

7351 

7440 

89 

7 

7529 

7618 

7707 

7796 

7886 

7975 

8064 

8153 

8242 

8331 

89 

8 

8420 

8509 

8598 

8687 

8776 

8865 

8953 

9042 

9131 

9220 

89 

9 

9309 

9398 

9486 

9575 

9664 

9753 

9841 

9930 

690019 

690107 

89 

490 

690196 

690285 

690373 

690462 

690550 

690639 

690728 

690816 

690905 

690993 

89 

1 

1081 

1170 

1258 

1347 

1435 

1524 

1612 

1700 

1789 

1877 

88 

2 

1965 

2053 

2142 

2230 

2318 

2406 

2494 

2583 

2671 

2759 

88 

3 

2847 

2935 

3023 

3111 

3199 

3287 

3375 

3403 

3551 

3639 

88 

4 

3727 

3815 

3903 

3991 

4078 

4166 

4254 

4342 

4430 

451-7 

88 

5 

4605 

4693 

4781 

4868 

4956 

5044 

5131 

5219 

5307 

5394 

88 

6 

5482 

5569 

5657 

5744 

5832 

5919 

6007 

6094 

6182 

6269 

87 

7 

6356 

6444 

6531 

6618 

6706 

6793 

6880 

6968 

7055 

7142 

87 

8 

7229 

7317 

7404 

7491 

7578 

7665 

7752 

7839 

7926 

8014 

87 

9 

8101 

8188 

8275 

8362 

8449 

8535 

8622 

8709 

8796 

8883 

87 

500 

698970 

699057 

699144 

699231 

[699317 

699404 

699491 

699578 

699064 

699751 

87 

1 

9838 

9924 

700011 

510098 

700184 

700271 

700358 

700444 

700531 

700617 

87 

2 

700704 

700790 

0871 

0963 

1050 

1136 

1222 

1309 

1395 

1482 

86 

3 

1568 

1654 

1741 

1827 

1913 

1999 

2086 

2172 

2258 

2344 

86 

4 

2431 

2517 

2603 

2689 

2775 

2861 

2947 

3033 

3119 

3205 

86 

5 

3291 

3377 

3463 

3549 

3635 

3721 

3807 

3893 

3979 

4065 

86 

6 

4151 

4236 

4322 

4408 

4494 

4579 

4665 

4751 

4837 

4922 

86 

7 

5008 

5094 

5179 

5265 

5350 

5436 

5522 

5607 

5693 

5778 

86 

8 

5864 

5949 

6035 

6120 

6206 

6291 

6376 

6462 

6547 

6632 

85 

9 

6718 

6803 

6888 

6974 

7059 

7144 

7229 

7315 

7400 

7485 

85 

510 

707570 

707655 

707740 

707826 

707911 

707996 

708081 

708166 

708251 

708336 

85 

1 

8421 

8506 

8591 

8676 

8761 

8846 

8931 

9015 

9100 

9185 

85 

2 

9270 

9355 

9440 

9524 

9609 

9694 

9779 

9863 

9948 

710033 

85 

3 

710117 

710202 

710287 

710371 

71045C 

710540 

710625 

710710 

710794 

0879 

85 

4 

0963 

1048 

1132 

1217 

1301 

1385 

1470 

1554 

1639 

1723 

84 

5 

1807 

1892 

1976 

2060 

2144 

2229 

2313 

2397 

2481 

2566 

84 

6 

2650 

2734 

2818 

2902 

2986 

3070 

3154 

3238 

3323 

3407 

84 

7 

3491 

3575 

3659 

3742 

3826 

3910 

3994 

4078 

4162 

4246 

84 

8 

4330 

4414 

4497 

4581 

4665 

4749 

4833 

4916 

5000 

5084 

84 

9 

5167 

5251 

5335 

5418 

5502 

5586 

5669 

5753 

5836 

5920 

84 

No. 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diff. 



















































TABLE I. LOGARITHMS OF NUMBERS. 9 


'No 

° 

1 

o 

3 

4 

5 

6 

7 

8 

9 

Diff. 

52C 

716003 

716087 

716170 

716254 

716337 

716421 

716504 

716588 

716671 

716754 

83 

1 

6838 

6921 

7004 

708? 

7171 

72.54 

7338 

7421 

7501 

7587 

83 

2 

7671 

7754 

7837 

7920 

8003 

8086 

8169 

8253 

8336 

8419 

83 

3 

8502 

8585 

8668 

8751 

8834 

8917 

9006 

9083 

9165 

9248 

83 

4 

9331 

9414 

9497 

9580 

9663 

9745 

9828 

9911 

9994 

720077 

83 

5 

720159 

720242 

720325 

720407 

720490 

720573 

720655 

720738 

720821 

0903 

83 

6 

0986 

1068 

1151 

1233 

1316 

1398 

1481 

1563 

1646 

1728 

82 

7 

1811 

1893 

1975 

2058 

2140 

2222 

2305 

2387 

2469 

2552 

82 

8 

2634 

2716 

2798 

2881 

2963 

3045 

3127 

3209 

3291 

3374 

82 

9 

3456 

3538 

3620 

3702 

3784 

3866 

3948 

4030 

4112 

4194 

82 

530 

724276 

724358 

724440 

724522 

724604 

724685 

724767 

724849 

724931 

725013 

82 

1 

5095 

5176 

5258 

5340 

5422 

5503 

5585 

5667 

5748 

5830 

82 

2 

5912 

5993 

6075 

6156 

6238 

6320 

6401 

6483 

6564 

6646 

82 

3 

6727 

6809 

6890 

6972 

7053 

7134 

7216 

7297 

7379 

7460 

81 

4 

7541 

7623 

7704 

7785 

7866 

7948 

8029 

8110 

8191 

8273 

81 

5 

8354 

8435 

8516 

8597 

8678 

8759 

8X41 

8922 

9003 

9084 

81 

6 

9165 

9246 

9327 

9408 

9489 

9570 

9651 

9732 

9813 

9893 

81 

7 

9974 

730055 

730136 

730217 

730298 

730378 

730459 

730540 

730621 

730702 

81 

8 

730782 

0863 

0944 

1024 

1105 

1186 

1266 

1347 

1428 

1508 

81 

9 

1589 

1669 

1750 

1830 

1911 

1991 

2072 

2152 

2233 

2313 

81 

540 

732394 

732474 

732555 

732635 

732715 

732796 

732876 

732956 

733037 

733117 

80 

1 

3197 

3278 

3358 

3438 

3518 

3598 

3679 

3759 

3839 

3919 

80 

2 

3999 

4079 

4160 

4240 

4320 

4400 

4480 

4560 

4640 

4720 

80 

3 

4800 

4880 

4960 

5040 

5120 

5200 

5279 

5359 

5439 

5519 

80 

4 

5599 

5679 

5759 

5838 

5918 

5998 

6078 

6157 

6237 

6317 

80 

5 

6397 

6476 

6556 

6635 

6715 

6795 

6874 

6954 

7034 

7113 

80 

6 

7193 

7272 

7352 

7431 

7511 

7590 

7670 

7749 

7829 

7908 

79 

7 

7987 

8067 

8146 

8225 

8305 

8384 

8463 

8.543 

8622 

8701 

79 

8 

8781 

8860 

8939 

9018 

9097 

9177 

9256 

9335 

9414 

9493 

79 

9 

9572 

9651 

9731 

9810 

9889 

9968 

740047 

740126 

740205 

740284 

79 

550 

740363 

740442 

740521 

740600 

740678 

740757 

740836 

740915 

740994 

741073 

79 

1 

1152 

1230 

1309 

1388 

1467 

1546 

1624 

1703 

1782 

1860 

79 

2 

1939 

2018 

2096 

2175 

2254 

2332 

2411 

2489 

2568 

2647 

79 

3 

2725 

2804 

2882 

2961 

3039 

3118 

3196 

3275 

3353 

3431 

78 

4 

3510 

3588 

3667 

3745 

3823 

3902 

3980 

4058 

4136 

4215 

78 

5 

4293 

4371 

4449 

4528 

4606 

4684 

4762 

4840 

4919 

4997 

78 

6 

5075 

5153 

5231 

5309 

5387 

5465 

5543 

5621 

5699 

5777 

78 

7 

5855 

5933 

6011 

6089 

6167 

6245 

6323 

6401 

6479 

6556 

78 

8 

6634 

6712 

6790 

6868 

6945 

7023 

7101 

7179 

7256 

7334 

78 

9 

7412 

7489 

7567 

7645 

7722 

7800 

7878 

7955 

8033 

8110 

78 

560 

748188 

748266 

748343 

748421 

748498 

748576 

748653 

748731 

748808 

748885 

77 

1, 

8963 

9040 

9118 

9195 

9272 

9350 

9427 

9504 

9582 

9659 

77 

2 

9736 

9814 

9891 

9968 

750045 

750123 

750200 

750277 

750354 

750431 

77 

3 

750508 

750586 

750663 

750740 

0817 

0894 

0971 

1048 

1125 

1202 

77 

4 

1279 

1356 

1433 

1510 

1587 

1664 

1741 

1818 

1895 

1972 

77 

5 

2048 

2125 

2202 

2279 

2356 

2433 

2509 

2586 

2663 

2740 

77 ’ 

G 

2816 

2893 

2970 

3047 

3123 

3200 

3277 

3353 

3430 

3506 

77 

7 

3583 

3660 

3736 

3813 

3889 

3966 

4042 

4119 

4195 

4272 

77 

8 

4348 

4425 

4501 

4578 

4654 

4730 

4807 

4883 

4960 

5036 

76 

9 

5112 

5189 

5265 

5341 

6417 

5494 

5570 

5646 

5722 

5799 

76 

570 

755875 

755951 

750027 

756103 

756180 

756256 

756332 

756408 

756484 

756560 

76 

1 

6636 

6712 

6788 

6864 

6940 

7016 

7092 

7168 

7244 

7320 

76 

2 

7396 

7472 

7548 

7624 

7700 

7775 

7851 

7927 

8003 

8079 

76 

3 

8155 

8230 

8306 

8382 

8458 

8533 

8609 

8685 

8761 

8836 

76 

4 

8912 

8988 

9063 

9139 

9214 

9290 

9366 

9441 

9517 

9592 

76 

5 

9668 

9743 

9819 

9894 

9970 

760045 

760121 

760196 

760272 

760347 

75 

6 

760422 

760498 

760573 

760649 

760724 

0799 

0875 

0950 

1025 

1101 

75 

7 

1176 

1251 

1326 

1402 

1477 

1552 

1627 

1702 

1778 

1853 

75 

8 

1928 

2003 

2078 

2153 

2228 

2303 

2378 

2453 

2529 

2604 

75 

9 

2679 

2754 

2829 

2904 

2978 

3053 

3128 

3203 

3278 

3353 

75 

No. 

0 

I 

o 

3 

4 

5 

6 

7 

8 

9 

Dlff. 


B 




















































































10 TABLE I. LOGARITHMS OF NUMBERS. 


No. 

0 

1 

2 

3 

4 

5 

<1 

7 

8 

9 

Diff 

580 

763428 

763503 

763578 

763653 

763727 

763802 

763877 

763952 

764027 

764101 

75 

1 

4176 

4251 

4326 

4400 

4475 

4550 

4624 

4699 

4774 

4848 

75 

2 

4923 

4998 

5072 

5147 

5221 

5296 

5370 

5445 

5520 

5594 

75 

3 

5669 

5743 

5818 

5892 

5966 

6041 

6115 

6190 

6264 

6338 

74 

4 

6413 

6487 

6562 

6636 

6710 

6785 

6859 

6933 

7007 

7082 

74 

5 

7156 

7230 

7304 

7379 

7453 

7527 

7601 

7675 

7749 

7823 

74 

6 

7898 

7972 

8046 

8120 

8194 

8268 

8342 

8416 

8490 

8564 

74 

7 

8638 

8712 

8786 

8860 

8934 

9008 

9082 

91.56 

9230 

9303 

74 

8 

9377 

9451 

9525 

9599 

9673 

9746 

9820 

9894 

9968 

770042 

74 

9 

770115 

770189 

770263 

770336 

770410 

770484 

770557 

770631 

770705 

0778 

74 

590 

770852 

770926 

770999 

771073 

771146 

771220 

771293 

771367 

771440 

771514 

74 

1 

1587 

1661 

1734 

1808 

1881 

1955 

2028 

2102 

2175 

2248 

73 

2 

2322 

2395 

2468 

2542 

2615 

2688 

2762 

2835 

2908 

2981 

73 

3 

3055 

3128 

3201 

3274 

3348 

3421 

3494 

3567 

3640 

3713 

73 

4 

3786 

3860 

3933 

4006 

4079 

4152 

4225 

4298 

4371 

4444 

73 

5 

4517 

4590 

4663 

4736 

4809 

4882 

4955 

5028 

5100 

5173 

73 

6 

5246 

5319 

5392 

5465 

5538 

5610 

5683 

5756 

5829 

5902 

73 

7 

5974 

6047 

6120 

6193 

6265 

6338 

6411 

6483 

6556 

6629 

73 

8 

6701 

6774 

6846 

6919 

' 6992 

7064 

7137 

7209 

7282 

7354 

73 

9 

7427 

7499 

7572 

7644 

7717 

7789 

78G2 

7934 

8006 

8079 

72 

GOO 

778151 

778224 

778296 

778368 

778441 

778513 

778585 

778658 

778730 

778802 

72 

1 

8874 

8947 

9019 

9091 

9163 

9236 

9308 

9380 

9452 

9524 

72 

2 

9596 

9669 

9741 

9813 

9885 

9957 

780029 

780101 

780173 

780245 

72 

3 

780317 

780389 

780461 

780533 

780605 

780677 

0749 

0821 

0893 

0965 

72 

4 

1037 

1109 

1181 

1253 

1324 

1396 

1468 

1540 

1612 

1684 

72 

5 

1755 

1827 

1899 

1971 

2042 

2114 

2186 

2258 

2329 

2401 

72 

6 

2473 

2544 

2616 

2688 

2759 

2831 

2902 

2974 

3046 

3117 

72 

7 

3189 

3260 

3332 

3403 

3475 

3546 

3618 

3689 

3761 

3832 

71 

8 

3904 

3975 

4046 

4118 

4189 

4261 

4332 

4403 

4475 

4546 

71 

9 

4617 

4689 

4760 

4831 

4902 

4974 

5045 

5116 

5187 

5259 

71 

610 

785330 

785401 

78.5472 

785543 

785615 

785686 

785757 

785828 

785899 

785970 

71 

1 

6041 

6112 

6183 

6254 

6325 

6396 

6467 

6538 

6609 

6680 

71 

2 

6751 

6822 

6893 

6964 

7035 

7106 

7177 

7248 

7319 

7390 

71 

3 

7460 

7531 

7602 

7673 

7744 

7815 

7885 

7956 

8027 

8098 

71 

4 

8168 

8239 

8310 

8381 

8451 

8522 

8593 

8663 

8734 

8804 

71 

5 

8875 

8946 

9016 

9087 

9157 

9228 

9299 

9369 

9440 

9510 

71 

G 

9581 

9651 

9722 

9792 

9863 

9933 

790004 

790074 

790144 

790215 

70 

7 

790285 

790356 

790426 

790496 

790567 

790637 

0707 

0778 

0848 

0918 

70 

8 

0988 

1059 

1129 

1199 

1269 

1340 

1410 

1480 

1550 

1620 

70 

9 

1691 

1761 

1831 

1901 

1971 

2041 

2111 

2181 

'2252 

2322 

70 

620 

792392 

792462 

792532 

792602 

792672 

792742 

792812 

792882 

792952 

793022 

70 

1 

3092 

3162 

3231 

3301 

3371 

3441 

3511 

3581 

3651 

3721 

70 

2 

3790 

3860 

3930 

4000 

4070 

4139 

4209 

4279 

4349 

4418 

70 

3 

4488 

4558 

4627 

4697 

4767 

4836 

4906 

4976 

5045 

5115 

70 

4 

5185 

5254 

5324 

5393 

5463 

5532 

5602 

5672 

5741 

5811 

70 

5 

5880 

5949 

6019 

6088 

6158 

6227 

6297 

6366 

6436 

6505 

69 

G 

6574 

6644 

6713 

6782 

6852 

6921 

6990 

7060 

7129 

7198 

69 

7 

7268 

7337 

7406 

7475 

7545 

7614 

7683 

7752 

7821 

7890 

69 

8 

7960 

8029 

8098 

8167 

8236 

8305 

8374 

8443 

8513 

8582 

69 

9 

8651 

8720 

8789 

8858 

8927 

8996 

9065 

9134 

9203 

9272 

69 

630 

799341 

799409 

799478 

799547 

799616 

799685 

799754 

799823 

799892 

799961 

69 

1 

800029 

800098 

800167 

800236 

800:505 

800373 

800442 

800511 

800580 

800648 

69 

2 

0717 

0786 

0854 

0923 

0992 

1061 

1129 

1198 

1266 

1335 

69 

3 

1404 

1472 

1541 

1609 

1678 

1747 

1815 

1884 

1952 

2021 

69 

4 

2089 

2158 

2226 

2295 

2363 

2432 

2500 

2568 

2637 

2705 

69 

5 

2774 

2842 

2910 

2979 

3047 

3116 

3184 

3252 

3321 

3389 

68 

6 

3457 

3525 

3594 

3662 

3730 

3798 

3867 

3935 

4003 

4071 

68 

7 

4139 

4208 

4276 

4344 

4412 

4480 

4548 

4616 

4685 

4753 

68 

8 

4821 

4889 

4957 

5025 

5093 

5161 

5229 

5297 

5365 

5433 

68 

9 

5501 

5569 

5637 

5705 

5773 

5841 

5908 

5976 

6044 

6112 

68 

IVo. 

0 

1 

o 

3 

4 

5 

6 

7 

8 

9 

Diff. 





























































table t. logarithms of numbers. 11 


No 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diff. 

640 

806180 

806248 

80631C 

806384 

8064511806511 

806587 

806655 

806723 

80679C 

68 

1 

6858 

6926 

6994 

7061 

7129 

7197 

7264 

7332 

7400 

7461 

68 

2 

7535 

7603 

7670 

7738 

7806 

7873 

7941 

8008 

8076 

8143 

68 

3 

8211 

8279 

8346 

8414 

8481 

8549 

8616 

8684 

8751 

8818 

67 

4 

8886 

8953 

9021 

9088 

9156 

9223 

9290 

9358 

9425 

9492 

67 

5 

9560 

9627 

9694 

9762 

9829 

9896 

9964 

810031 

810098 

810165 

67 

6 810233 

810300 

810367 

810434 

810501 

810569 

810636 

0703 

0770 

0837 

67 

7 

0904 

0971 

1039 

1106 

1173 

1240 

1307 

1374 

1441 

1508 

67 

8 

1575 

1642 

1709 

1776 

1843 

1910 

1977 

2044 

2111 

2178 

67 

9 

2245 

2312 

2379 

2445 

2512 

2579 

2646 

2713 

2780 

2847 

67 

650 

812913 

812980 

813047 

813114 

813181 

813247 

813314 

813381 

813448 

813514 

67 

1 

3581 

3648 

3714 

3781 

3848 

3914 

3981 

4048 

4114 

4181 

67 

2 

4248 

4314 

4381 

4447 

4514 

4581 

4647 

4714 

4780 

4847 

67 

3 

4913 

4980 

5046 

5113 

5179 

5246 

5312 

5378 

5445 

6511 

66 

4 

5578 

5644 

5711 

5777 

5843 

5910 

5976 

6042 

6109 

6175 

66 

5 

6241 

6308 

6374 

6440 

6506 

6573 

6639 

6705 

6771 

6838 

66 

6 

6904 

6970 

7036 

7102 

7169 

7235 

7301 

7367 

7433 

7499 

66 

7 

7565 

7631 

7698 

7764 

7830 

7896 

7962 

8028 

8094 

8160 

66 

8 

8226 

8292 

8358 

8424 

8490 

8556 

8622 

8688 

8754 

8820 

66 

9 

8885 

8951 

9017 

9083 

9149 

9215 

9281 

9346 

9412 

9478 

66 

660 

819544 

819610 

819676 

819741 

819807 

819873 

819939 

820004 

820070 

820136 

66 

1 

820201 

820267 

820333 

820399 

820464 

820530 

820595 

0661 

0727 

0792 

66 

2 

0858 

0924 

0989 

1055 

1120 

1186 

1251 

1317 

1382 

1448 

66 

3 

1514 

1579 

1645 

1710 

1775 

1841 

1906 

1972 

2037 

2103 

66 

4 

2168 

2233 

2299 

2364 

2430 

2495 

2560 

2626 

2691 

2756 

65 

5 

2822 

2887 

2952 

3018 

3083 

3148 

3213 

3279 

3344 

3409 

65 

6 

3474 

3539 

3605 

3670 

3735 

3800 

3865 

3930 

3996 

4061 

65 

7 

4126 

4191 

4256 

4321 

4386 

4451 

4516 

4581 

4646 

4711 

65 

8 

4776 

4841 

4906 

4971 

5036 

5101 

5166 

5231 

5296 

5361 

65 

9 

5426 

5491 

5556 

5621 

5686 

5751 

5815 

5880 

5945 

6010 

65 

670 

826075 

826140 

826204 

826269 

826334 

826399 

826464 

826528 

826593 

826658 

65 

1 

6723 

6787 

6852 

6917 

6981 

7046 

7111 

7175 

7240 

7305 

65 

2 

7369 

7434 

7499 

7563 

7628 

7692 

7757 

7821 

7886 

7951 

65 

3 

8015 

8080 

8144 

8209 

8273 

8338 

8402 

8467 

8531 

8595 

64 

4 

8660 

8724 

8789 

8853 

8918 

8982 

9046 

9111 

9175 

9239 

64 

5 

9304 

9368 

9432 

9497 

9561 

9625 

9690 

9754 

9818 

9882 

64 

6 

9947 

830011 

830075 

830139 

830204 

830268 

830332 

830396 

830460 

830525 

64 

7 

830589 

0653 

0717 

0781 

0845 

0909 

0973 

1037 

1102 

1166 

64 

8 

1230 

1294 

1358 

1422 

I486 

1550 

1614 

1678 

1742 

1806 

64 

9 

1870 

1934 

1998 

2062 

2126 

2189 

2253 

2317 

2381 

2445 

64 

680 

832509 

832573 

832637 

832700 

832764 

832828 

832892 

832956 

833020 

833083 

64 

1 

3147 

3211 

3275 

3338 

3402 

3466 

3530 

3593 

3657 

3721 

64 

2 

3784 

3848 

3912 

3975 

4039 

4103 

4166 

4230 

4294 

4357 

64 

3 

4421 

4484 

4548 

4611 

4675 

4739 

4802 

4866 

4929 

4993 

64 

4 

5056 

5120 

5183 

5247 

5310 

5373 

5437 

5500 

5564 

5627 

63 

5 

5691 

5754 

5817 

5881 

5944 

6007 

6071 

6134 

6197 

6261 

63 

6 

6324 

6387 

6451 

6514 

6577 

6641 

6704 

6767 

6830 

6894 

63 

7 

6957 

7020 

7083 

7146 

7210 

7273 

7336 

7399 

7462 

7525 

63 

8 

7588 

7652 

7715 

7778 

7841 

7904 

7967 

8030 

8093 

8156 

63 

9 

8219 

8282 

8345 

8408 

8471 

8534 

8597 

8660 

8723 

8786 

63 

690 

838849 

838912 

838975 

839038 

839101 

839164 

839227 

839289 

839352 

839415 

63 

1 

9478 

9541 

9604 

9667 

9729 

9792 

9855 

9918 

9981 

840043 

63 

2 

840106 

840169 

840232 

840294 

840357 

840420 

840482 

840545 

840608 

0671 

63 

3 

0733 

0796 

0859 

0921 

0984 

1046 

1109 

1172 

1234 

1297 

63 

4 

1359 

1422 

1485 

1547 

1610 

1672 

1735 

1797 

1860 

1922 

63 

5 

1985 

2047 

2110 

2172 

2235 

2297 

2360 

2422 

2484 

2547 

62 

6 

2609 

2672 

2734 

2796 

2859 

2921 

2983 

3046 

3108 

3170 

62 

7 

3233 

3295 

3357 

3420 

3482 

3544 

3606 

3669 

3731 

3793 

62 

8 

3855 

3918 

3980 

4042 

4104 

4166 

4229 

4291 

4353 

4415 

62 

9 

4477 

4539 

4601 

4664 

4726 

4788 

4850 

4912 

4974 

5036 

62 

No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diff. 


















































































12 TABLE I. LOGARITHMS OF NUMBERS. 


No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diff. 

700 

845098 

845160 

845222 

845284 

845346 

845408 

845470 

845532 

845594 

845656 

62 

1 

5718 

5780 

5842 

5904 

5966 

6028 

6090 

6151 

6213 

6275 

62 

2 

6337 

6399 

6461 

6523 

6585 

6646 

6708 

6770 

6832 

6894 

62 

3 

6955 

7017 

7079 

7141 

7202 

7264 

7326 

7388 

7449 

7511 

62 

4 

7573 

7634 

7696 

7758 

7819 

7881 

7943 

8004 

8066 

8128 

62 

5 

8189 

8251 

8312 

8374 

8435 

8497 

8559 

8620 

8682 

8743 

62 

6 

8805 

8866 

8928 

8989 

9051 

9112 

9174 

9235 

9297 

9358 

6L 

7 

9419 

9481 

9542 

9604 

9665 

9726 

9788 

9849 

9911 

9972 

61 

8 

850033 

850095 

850156 

850217 

850279 

850340 

850401 

850462 

850524 

850585 

61 

9 

0646 

0707 

0769 

0830 

0891 

0952 

1014 

1075 

1136 

1197 

61 

710 

851258 

851320 

851381 

851442 

851.503 

851564 

851625 

851686 

851747 

851809 

61 

1 

1870 

1931 

1992 

2053 

2114 

2175 

2236 

2297 

2358 

2419 

61 

2 

2480 

2541 

2602 

2663 

2724 

2785 

2846 

2907 

2968 

3029 

61 

3 

3090 

3150 

3211 

3272 

3333 

3394 

3455 

3516 

3577 

3637 

61 

4 

3698 

3759 

3820 

3881 

3941 

4002 

4063 

4124 

4185 

4245 

61 

5 

4306 

4367 

4428 

4488 

4549 

4610 

4670 

4731 

4792 

4852 

61 

6 

4913 

4974 

5034 

5095 

5156 

5216 

5277 

5337 

5398 

5459 

61 

7 

5519 

5580 

5640 

5701 

5761 

5822 

5882 

5943 

6003 

6064 

61 

8 

6124 

6185 

6245 

6306 

6366 

6427 

6487 

6548 

6608 

6668 

60 

9 

6729 

6789 

6850 

6910 

6970 

7031 

7091 

7152 

7212 

7272 

60 

720 

857332 

857393 

857453 

857513 

857574 

857634 

857694 

857755 

857815 

857875 

60 

1 

7935 

7995 

8056 

8116 

8176 

8236 

8297 

8357 

8417 

8477 

60 

2 

8537 

8597 

8657 

8718 

8778 

8838 

8898 

8958 

9018 

9078 

60 

3 

9138 

9198 

9258 

9318 

9379 

9439 

9499 

9559 

9619 

9679 

60 

4 

9739 

9799 

9859 

9918 

9978 

860038 

860098 

860158 

860218 

860278 

60 

5 

860338 

860398 

860458 

860518 

860578 

0637 

0697 

0757 

0817 

0877 

60 

0 

0937 

0996 

1056 

1116 

1176 

1236 

1295 

1355 

1415 

1475 

60 

7 

1534 

1594 

1654 

1714 

1773 

1833 

1893 

1952 

2012 

2072 

60 

8 

2131 

2191 

2251 

2310 

2370 

2430 

2489 

2549 

2608 

2668 

60 

9 

2728 

2787 

2847 

2906 

2966 

3025 

3085 

3144 

3204 

3263 

60 

730 

863323 

863382 

863442 

863501 

863561 

863620 

863680 

863739 

863799 

863858 

59 

1 

3917 

3977 

4036 

4096 

4155 

4214 

4274 

4333 

4392 

4452 

59 

2 

4511 

4570 

4630 

4689 

4748 

4808 

4867 

4926 

4985 

5045 

59 

3 

5104 

5163 

5222 

5282 

5341 

5400 

5459 

5519 

5578 

5637 

59 

4 

5696 

5755 

5814 

5874 

5933 

5992 

6051 

6110 

6169 

6228 

59 

5 

6287 

6346 

6405 

6465 

6524 

6583 

6642 

6701 

6760 

6819 

59 

6 

6878 

6937 

6996 

7055 

7114 

7173 

7232 

7291 

7350 

7409 

59 

7 

7467 

7526 

7585 

7644 

7703 

7762 

7821 

7880 

7939 

7998 

59 

8 

8056 

8115 

8174 

8233 

8292 

8350 

8409 

8468 

8527 

8586 

59 

9 

8644 

8703 

8762 

8821 

8879 

8938 

8997 

9056 

9114 

9173 

59 

740 

869232 

869290 

869349 

869408 

869466 

869525 

869584 

869642 

869701 

869760 

59 

1 

9818 

9877 

9935 

9994 

870053 

870111 

870170 

870228 

870287 

870345 

59 

2 

870404 

870462 

870521 

870579 

0638 

0696 

0755 

0813 

0872 

0930 

58 

3 

0989 

1047 

1106 

1164 

1223 

1281 

1339 

1398 

1456 

1515 

58 

4 

1573 

1631 

1690 

1748 

1806 

1865 

1923 

1981 

2040 

2098 

58 

5 

2156 

2215 

2273 

2331 

2389 

2448 

2506 

2564 

2622 

2681 

58 

6 

2739 

2797 

2855 

2913 

2972 

3030 

3088 

3146 

3204 

3262 

58 

rr 

i 

3321 

3379 

3437 

3495 

3553 

3611 

3669 

3727 

3785 

3844 

58 

8 

3902 

3960 

4018 

4076 

4134 

4192 

4250 

4308 

4366 

4424 

58 

9 

4482 

4540 

4598 

4656 

4714 

4772 

4830 

4888 

4945 

5003 

58 

750 

875061 

875119 

875177 

875235 

875293 

875351 

875409 

875466 

875524 

875582 

58 

1 

5640 

5698 

5756 

5813 

5871 

5929 

5987 

6045 

6102 

6160 

58 

2 

6218 

6276 

6333 

6391 

6449 

6507 

6564 

6622 

6680 

6737 

58 

3 

6795 

6853 

6910 

6968 

7026 

7083 

7141 

7199 

7256 

7314 

58 

4 

7371 

7429 

7487 

7544 

7602 

7659 

7717 

7774 

7832 

7889 

58 

5 

7947 

8004 

8062 

8119 

8177 

8234 

8292 

8349 

8407 

8464 

57 

G 

8522 

8579 

8637 

8694 

8752 

8809 

8866 

8924 

8981 

9039 

57 

7 

9096 

9153 

9211 

9268 

9325 

9383 

9440 

9497 

9555 

9612 

57 

8 

9669 

9726 

9784 

9841 

9898 

9956 

880013 

880070 

880127 

880185 

57 

9 

880242 

880299 

8803.56 

880413 

880471 

880528 

0585 

0642 

0699 

0756 

57 

No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diff. 










































TABLE I. LOGARITHMS OF NUMBERS, 


IB 


No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diff. 

760 

880814 

880871 

880928 

880985 

881042 881099 

881156 

881213 

881271 

881328 

57 

1 

1385 

1442 

1499 

1556 

1613 

1670 

1727 

1784 

1841 

1898 

57 

2 

1955 

2012 

2069 

2126 

2183 

2240 

2297 

2354 

2411 

2468 

57 

3 

2525 

2581 

2638 

2695 

2752 

2809 

2866 

2923 

2980 

3037 

57 

4 

3093 

3150 

3207 

3264 

3321 

3377 

3434 

3491 

3548 

3605 

57 

5 

3661 

3718 

3775 

3832 

3888 

3945 

4002 

4059 

4115 

4172 

57 

6 

4229 

4285 

4342 

4399 

4455 

4512 

4569 

4625 

4682 

4739 

57 

7 

4795 

4852 

4909 

4965 

5022 

5078 

5135 

5192 

5248 

5305 

57 

8 

5361 

5418 

5474 

5531 

5587 

5644 

5700 

5757 

5813 

5870 

57 

9 

5926 

5983 

6039 

6096 

6152 

6209 

6265 

6321 

6378 

6434 

56 

770 

886491 

886547 

886604 

886660 

886716 

886773 

886829 

886885 

886942 

886998 

56 

1 

7054 

7111 

7167 

7223 

7280 

7336 

7392 

7449 

7505 

7561 

56 

2 

7617 

7674 

7730 

7786 

7842 

7898 

7955 

8011 

8067 

8123 

56 

3 

8179 

8236 

8292 

8348 

8404 

8460 

8516 

8573 

8629 

8685 

56 

4 

8741 

8797 

8853 

8909 

8965 

9021 

9077 

9134 

9190 

9246 

56 

5 

9302 

9358 

9414 

9470 

9526 

9582 

9638 

9694 

9750 

9806 

56 

6 

9862 

9918 

9974 

890030 

890086 

890141 

890197 

890253 

890309 

890365 

56 

7 

890421 

890477 

890533 

0589 

0645 

0700 

0756 

0812 

0868 

0924 

56 

8 

0980 

1035 

1091 

1147 

1203 

1259 

1314 

1370 

1426 

1482 

56 

9 

1537 

1593 

1649 

1705 

1760 

1816 

1872 

1928 

1983 

2039 

56 

780 

892095 

892150 

892206 

892262 

892317 

892373 

892429 

892484 

892540 

892595 

56 

1 

2651 

2707 

2762 

2818 

2873 

2929 

2985 

3040 

3096 

3151 

56 

2 

3207 

3262 

3318 

3373 

3429 

3484 

3540 

3595 

3651 

3706 

56 

3 

3762 

3817 

3873 

3928 

3984 

4039 

4094 

4150 

4205 

4261 

55 

4 

4316 

4371 

4427 

4482 

45.38 

4593 

4648 

4704 

4759 

4814 

55 

5 

4870 

4925 

4980 

5036 

5091 

5146 

5201 

5257 

5312 

5367 

55 

6 

5423 

5478 

5533 

5588 

5644 

5699 

5754 

5809 

5864 

5920 

55 

7 

5975 

6030 

6085 

6140 

6195 

6251 

6306 

6361 

6416 

6471 

55 

8 

6526 

6581 

6636 

6692 

6747 

6802 

6857 

6912 

6967 

7022 

55 

9 

7077 

7132 

7187 

7242 

7297 

7352 

7407 

7462 

7517 

7572 

55 

790 

897627 

897682 

897737 

897792 

897847 

897902 

897957 

898012 

898067 

898122 

55 

1 

8176 

8231 

8286 

8341 

8396 

8451 

8506 

8561 

8615 

8670 

55 

2 

8725 

8780 

8835 

8890 

8944 

8999 

9054 

9109 

9164 

9218 

55 

3 

9273 

9328 

9383 

9437 

9492 

9547 

9602 

9656 

9711 

9766 

55 

4 

9821 

9875 

9930 

9985 

9000.39 

900094 

900149 

900203 

900258 

900312 

55 

5 

900367 

900422 

900476 

900531 

0586 

0640 

0695 

0749 

0804 

0859 

55 

6 

0913 

0968 

1022 

1077 

1131 

1186 

1240 

1295 

1349 

1404 

55 

7 

1458 

1513 

1567 

1622 

1676 

1731 

1785 

1840 

1894 

1948 

&4 

8 

2003 

2057 

2112 

2166 

2221 

2275 

2329 

2384 

2438 

2492 

54 

9 

2547 

2601 

2655 

2710 

2764 

2818 

2873 

2927 

2981 

3036 

54 

800 

903090 

903144 

903199 

903253 

903307 

903361 

903416 

903470 

903524 

903578 

54 

1 

3633 

3687 

3741 

3795 

3849 

3904 

3958 

4012 

4066 

4120 

54 

2 

4174 

4229 

4283 

4337 

4391 

4445 

4499 

4553 

4607 

4661 

54 

3 

4716 

4770 

4824 

4878 

4932 

4986 

5040 

5094 

5148 

5202 

54 

4 

5256 

5310 

5364 

5418 

5472 

5526 

5580 

5634 

5688 

5742 

54 

5 

5796 

5850 

5904 

5958 

6012 

6066 

6119 

6173 

6227 

6281 

54 

6 

6335 

6389 

6443 

6497 

6551 

6604 

6658 

6712 

6766 

6820 

54 

7 

6874 

6927 

6981 

7035 

7089 

7143 

7196 

7250 

7304 

7358 

54 

8 

7411 

7465 

7519 

7573 

7626 

7680 

7734 

7787 

7841 

7895 

54 

9 

7949 

8002 

8056 

8110 

8163 

8217 

8270 

8324 

8378 

8431 

54 

810 

908485 

908539 

908592 

908646 

908699 

908753 

908807 

908860 

908914 

908967 

54 

1 

9021 

9074 

9128 

9181 

9235 

9289 

9342 

9396 

9449 

9503 

54 

2 

95.56 

9610 

9663 

9716 

9770 

9823 

9877 

9930 

9984 

910037 

53 

3 

910091 

910144 

910197 

910251 

910.304 

910.358 

910411 

910464 

910518 

0571 

53 

4 

0624 

0678 

0731 

0784 

0838 

0891 

0944 

0998 

1051 

1104 

53 

5 

1158 

1211 

1264 

1317 

1371 

1424 

1477 

1530 

1584 

1637 

53 

6 

1690 

1743 

1797 

18.50 

1903 

1956 

2009 

2063 

2116 

2169 

53 

7 

2222 

2275 

2328 

2381 

2435 

2488 

2541 

2594 

2647 

2700 

53 

8 

27.53 

2806 

2859 

2913 

2966 

3019 

3072 

3125 

3178 

3231 

53 

9 

3284 

3337 

3390 

3443 

3496 

3549 

3602 

3655 

3708 

3761 

53 

No. 

O 

1 

*> 

3 

4 | 

5 

6 

7 

8 

9 

Diff. 





















































TABLE I. LOGARITHMS OF NUMBERS, 


14 


No. 

0 

1 

o 

• » 

1 

5 

6 

7 

8 

9 

Kff. 

820 

913814 

913867 

913920 

913973 

914026 

914079 

914132 

914184 

914237 914290 

53 

1 

4343 

4396 

4449 

4502 

4555 

4608 

4660 

4713 

4766 

4819 

53 

2 

4872 

4925 

4977 

5030 

5083 

5136 

5189 

5241 

5294 

5347 

53 

3 

5400 

5453 

5505 

5558 

5611 

5664 

5716 

5769 

5822 

5875 

53 

4 

5927 

5980 

6033 

6085 

6138 

6191 

6243 

6296 

6349 

6401 

53 

5 

6454 

6.507 

6559 

6612 

6664 

6717 

6770 

6822 

6875 

6927 

53 

6 

6980 

7033 

7085 

7138 

7190 

7243 

7295 

7348 

7400 

7453 

53 

7 

7506 

7558 

7611 

7663 

7716 

7768 

7820 

7873 

7925 

7978 

52 

8 

8030 

8083 

8135 

8188 

8240 

8293 

8345 

8397 

8450 

8502 

52 

9 

8555 

8607 

8659 

8712 

8764 

8816 

8869 

8921 

8973 

9026 

52 

830 

919078 

919130 

919183 

919235 

919287 

919340 

919392 

919444 

919496 

919549 

52 

1 

9601 

9653 

9706 

9758 

9810 

9862 

9914 

9967 

9200191920071 

52 

2 

920123 

920176 

920228 

920280 

920332 

920384 

920436 

920489 

0541 

0593 

52 

3 

0645 

0697 

0749 

0801 

0853 

0906 

0958 

1010 

1062 

1114 

52 

4 

1166 

1218 

1270 

1322 

1374 

1426 

1478 

1530 

1582 

1634 

52 

5 

1686 

1738 

1790 

1842 

1894 

1946 

1998 

2050 

2102 

2154 

52 

G 

2206 

2258 

2310 

2362 

2414 

2466 

2518 

2570 

2622 

2674 

52 

7 

2725 

2777 

2829 

2881 

2933 

2985 

3037 

3089 

3140 

3192 

52 

8 

3244 

3296 

3348 

3399 

3451 

3503 

3555 

3607 

3658 

3710 

52 

9 

3762 

3814 

3865 

3917 

3969 

4021 

4072 

4124 

4176 

4228 

52 

840 

924279 

924331 

924383 

924434 

924486 

924538 

924589 

924641 

924693 

924744 

52 

1 

4796 

4848 

4899 

4951 

5003 

5054 

5106 

5157 

5209 

5261 

52 

2 

6312 

5364 

5415 

5467 

5518 

5570 

5621 

5673 

5725 

5776 

52 

3 

5828 

5879 

5931 

5982 

6034 

6085 

6137 

6188 

6240 

6291 

51 

4 

6342 

6394 

6445 

6497 

6548 

6600 

6651 

6702 

6754 

6805 

51 

5 

6857 

6908 

6959 

7011 

7062 

7114 

7165 

7216 

7268 

7319 

51 

G 

7370 

7422 

7473 

7524 

7576 

7627 

7678 

7730 

7781 

7832 

51 

7 

7883 

7935 

7986 

8037 

8088 

8140 

8191 

8242 

8293 

8345 

51 

8 

8396 

8447 

8498 

8549 

8601 

8652 

8703 

8754 

8805 

8857 

51 

9 

8908 

8959 

9010 

9061 

9112 

9163 

9215 

9266 

9317 

9368 

51 

850 

929419 

929470 

929521 

929572 

929623 

929674 

929725 

929776 

929827 

929879 

51 

1 

9930 

9981 

930032 

930083 

930134 

930185 

930236 

930287 

930338 

930389 

51 

2 

930440 

930491 

0542 

0592 

0643 

0694 

0745 

0796 

0847 

0898 

51 

3 

0949 

1000 

1051 

1102 

1153 

1204 

1254 

1305 

1356 

1407 

51 

4 

1458 

1509 

1560 

1610 

1661 

1712 

1763 

1814 

1865 

1915 

51 

5 

1966 

2017 

2068 

2118 

2169 

2220 

2271 

2322 

2372 

2423 

51 

G 

2474 

2524 

2575 

2626 

2677 

2727 

2778 

2829 

2879 

2930 

51 

7 

2981 

3031 

3082 

3133 

3183 

3234 

3285 

3335 

3386 

3437 

51 

8 

3487 

3538 

3589 

3639 

3690 

3740 

3791 

3841 

3892 

3943 

51 

9 

3993 

4044 

4091 

4145 

4195 

4246 

4296 

4347 

4397 

4448 

51 

8G0 

934498 

934549 

934599 

934650 

934700 

934751 

934801 

934852 

934902 

934953 

50 

1 

5003 

5054 

5104 

51.54 

5205 

5255 

5306 

5356 

5406 

5457 

50 

2 

5507 

5558 

5608 

5658 

5709 

5759 

5809 

5860 

5910 

5960 

50 

3 

6011 

6061 

6111 

6162 

6212 

6262 

6313 

6363 

6413 

6463 

50 

4 

6514 

6.564 

6614 

6665 

6715 

6765 

6815 

6865 

6916 

6966 

50 

5 

7016 

7066 

7117 

7167 

7217 

7267 

7317 

7367 

7418 

7468 

50 

6 

7518 

7568 

7618 

7668 

7718 

7769 

7819 

7869 

7919 

7969 

50 

7 

8019 

8069 

8119 

8169 

8219 

8269 

8320 

8370 

8420 

8470 

50 

8 

8520 

8570 

8620 

8670 

8720 

8770 

8820 

8870 

8920 

8970 

50 

9 

9020 

9070 

9120 

9170 

9220 

9270 

9320 

9369 

9419 

9469 

50 

870 

939519 

939569 

939619 

939669 

939719 

939769 

939819 

939869 

939918 

939968 

50 

1 

940018 

940068 

940118 

940168 

940218 

940267 

940317 

940367 

940417 

940467 

50 

2 

0516 

0566 

0616 

0666 

0716 

0765 

0815 

0865 

0915 

0964 

50 

3 

1014 

1064 

1114 

1163 

1213 

1263 

1313 

1362 

1412 

1462 

50 

4 

1511 

1561 

1611 

1660 

1710 

1760 

1809 

1859 

1909 

1958 

50 

5 

2008 

2058 

2107 

2157 

2207 

2256 

2306 

2355 

2405 

2455 

50 

6 

2.504 

2554 

2603 

2653 

2702 

2752 

2801 

2851 

2901 

2950 

50 

7 

3000 

3049 

3099 

3148 

3198 

3247 

3297 

3346 

3396 

3445 

49 

8 

3495 

3544 

3593 

3643 

3692 

3742 

3791 

3841 

3890 

3939 

49 

9 

3989 

4038 

4088 

4137 

4186 

4236 

4285 

4335 

4384 

4433 

49 

No. 

O 

1 

2 

3 

4 

5 

« 

7 

8 

9 

Diff. 







































































TABLE I. LOGARITHMS OF NUMBERS. 15 


No. 

0 

1 

o 

3 

4 

5 

6 

7 

8 

9 

Diff. 

880 

944483 

944532 

944581 

944631 

944680 

944729 

944779 

944828 

944877 

944927 

49 

1 

4976 

5025 

5074 

5124 

5173 

5222 

5272 

5321 

5370 

5419 

49 

2 

54G9 

5518 

5567 

5616 

5665 

5715 

5764 

5813 

5862 

5912 

49 

3 

59G1 

6010 

6059 

6108 

6157 

6207 

6256 

6305 

6354 

6403 

49 

4 

6452 

6501 

6551 

6600 

6649 

6698 

6747 

6796 

6845 

6894 

49 

5 

G943 

6992 

7041 

7090 

7140 

7189 

7238 

7287 

7336 

7385 

49 

6 

7434 

7483 

7532 

7581 

7630 

7679 

7728 

7777 

7826 

7875 

49 

7 

7924 

7973 

8022 

8070 

8119 

8168 

8217 

8266 

8315 

8364 

49 

8 

8413 

8462 

8511 

8560 

8609 

8657 

8706 

8755 

8804 

8853 

49 

9 

8902 

8951 

8999 

9048 

9097 

9146 

9195 

9244 

9292 

9341 

49 

890 

949390 

949439 

949488 

949536 

949585 

949634 

949683 

949731 

949780 

949829 

49 

1 

9878 

9926 

9975 

950024 

950073 

950121 

950170 

950219 

950267 

950316 

49 

2 

950365 

950414 

950462 

0511 

0560 

0008 

0657 

0706 

0754 

0803 

49 

3 

0851 

0900 

0949 

0997 

1046 

1095 

1143 

1192 

4240 

1289 

49 

4 

1338 

1386 

1435 

1483 

1532 

1580 

1629 

1677 

1726 

1775 

49 

5 

1823 

1872 

1920 

1969 

2017 

2066 

2114 

2163 

2211 

2260 

48 

6 

2308 

2356 

2405 

2453 

2502 

2550 

2599 

2647 

2696 

2744 

48 

7 

2792 

2841 

2889 

2938 

2986 

3034 

3083 

3131 

3180 

3228 

48 

8 

327G 

3325 

3373 

3421 

3470 

3518 

3566 

3615 

3663 

3711 

48 

9 

3760 

3808 

3856 

3905 

3953 

4001 

4049 

4098 

4146 

4194 

48 

900 

954243 

954291 

954339 

954387 

954435 

954484 

954532 

954580 

954628 

954677 

48 

1 

4725 

4773 

4821 

4869 

4918 

4966 

5014 

5062 

5110 

5158 

48 

2 

5207 

5255 

5303 

5351 

5399 

5447 

5495 

5543 

5592 

5640 

48 

3 

5688 

5736 

5784 

5832 

5880 

5928 

5976 

6024 

6072 

6120 

48 

4 

6168 

6216 

6265 

6313 

6361 

6409 

6457 

6505 

6553 

6601 

48 

5 

6649 

6697 

6745 

6793 

6840 

6888 

6936 

6984 

7032 

7080 

48 

G 

7128 

7176 

7224 

7272 

7320 

7368 

7416 

7464 

7512 

7559 

48 

r* 

i 

7607 

7655 

7703 

7751 

7799 

7847 

7894 

7942 

7990 

8038 

48 

8 

8086 

8134 

8181 

8229 

8277 

8325 

8373 

8421 

8468 

8516 

48 

9 

8564 

8612 

8659 

8707 

8755 

8803 

8850 

8898 

8946 

8994 

48 

910 

959041 

959089 

959137 

959185 

959232 

959280 

959328 

959375 

959423 

959471 

48 

1 

9518 

9566 

9614 

9661 

9709 

9757 

9804 

9852 

9900 

9947 

48 

2 

9995 

960042 

960090 

960138 

960185 

960233 

960280 

960328 

960376 

960423 

48 

3 

960471 

0518 

0566 

0613 

0661 

0709 

0756 

0804 

0851 

0899 

48. 

4 

0946 

0994 

1041 

1089 

1136 

1184 

1231 

1279 

1326 

1374 

47 

5 

1421 

1469 

1516 

1563 

1611 

1658 

1706 

1753 

1801 

1848 

47 

G 

1895 

1943 

1990 

2038 

2085 

2132 

2180 

2227 

2275 

2322 

47 

7 

2369 

2417 

2464 

2511 

2559 

2606 

2653 

2701 

2748 

2795 

47 

8 

2843 

2890 

2937 

2985 

3032 

3079 

3126 

3174 

3221 

3268 

47 

9 

3316 

3363 

3410 

3457 

3504 

3552 

3599 

3646 

3693 

3741 

47 

920 

963788 

963835 

963882 

963929 

963977 

964024 

964071 

964118 

964165 

964212 

47 

1 

4260 

4307 

4354 

4401 

4448 

4495 

4542 

4590 

4637 

4684 

47 

2 

4731 

4778 

4825 

4872 

4919 

4966 

5013 

5061 

5108 

5155 

47 

3 

5202 

5249 

5296 

5343 

5390 

5437 

5484 

5531 

5578 

5625 

47 

4 

5672 

5719 

5766 

5813 

5860 

5907 

5954 

6001 

6048 

6095 

47 

5 

6142 

6189 

6236 

6283 

6329 

6376 

6423 

6470 

6517 

6564 

47 

G 

6611 

6658 

6705 

6752 

6799 

6845 

6892 

6939 

6986 

7033 

47 

7 

7080 

7127 

7173 

7220 

7267 

7314 

7361 

7408 

7454 

7501 

47 

8 

7548 

7595 

7642 

7688 

77.35 

7782 

7829 

7875 

7922 

7969 

47 

9 

8016 

8062 

8109 

8156 

8203 

8249 

8296 

8343 

8390 

8436 

47 

930 

968483 

968530 

968576 

968623 

968670 

968716 

968763 

968810 

968856 

968903 

47 

1 

8950 

8996 

9043 

9090 

9136 

9183 

9229 

9276 

9323 

9369 

47 

2 

9416 

9463 

9509 

9556 

9602 

9649 

9695 

9742 

9789 

9835 

47 

3 

9882 

9928 

9975 

970021 

970068 

970114 

970161 

970207 

970254 

970300 

47 

4 

970347 

970393 

970440 

' 0486 

0533 

0579 

0626 

0672 

0719 

0765 

46 

5 

0812 

0858 

0904 

0951 

0997 

1044 

1090 

1137 

1183 

1229 

46 

6 

1276 

1322 

1369 

1415 

1461 

1508 

1554 

1601 

1647 

1693 

46 

7 

1740 

1786 

1832 

1879 

1925 

1971 

2018 

20(11 

2110 

2157 

46 

8 

2203 

2249 

2295 

2342 

2388 

2434 

2481 

2527 

2573 

2619 

46 

9 

2666 

2712 

2758 

2804 

2851 

2897 

2943 

2989 

3035 

3082 

46 

No- 

O 

1 

2 

3 

4 

5 1 

6 1 

7 

8 

9 

Diff. 
























































































16 TABLE I. LOGARITHMS OF NUMBERS. 


No. 

0 

1 

o 

At 

3 

4 

5 

6 

7 

8 

9 

Diff. 

940 

973128 

973174 

973220 

973266 

973313 

973359 

973405 

973451 

973497 

973.543 

46 

1 

3590 

3636 

3682 

3728 

3774 

3820 

3866 

3913 

3959 

4005 

46 

2 

4051 

4097 

4143 

4189 

4235 

4281 

4327 

4374 

4420 

4466 

46 

3 

4512 

4558 

4604 

4650 

4696 

4742 

4788 

4834 

4880 

4926 

46 

4 

4972 

5018 

5064 

5110 

5156 

5202 

5248 

5294 

5340 

5386 

46 

5 

5432 

5478 

5524 

5570 

5616 

5662 

5707 

5753 

5799 

5845 

46 

6 

5891 

5937 

5983 

6029 

6075 

6121 

6167 

6212 

6258 

6304 

46 

7 

6350 

6396 

6442 

6488 

6533 

6579 

6625 

6671 

6717 

6763 

46 

8 

6808 

6854 

6900 

6946 

6992 

7037 

7083 

7129 

7175 

7220 

46 

9 

7266 

7312 

7358 

7403 

7449 

7495 

7541 

7586 

7632 

7678 

46 

950 

977724 

977769 

977815 

977861 

977906 

977952 

977998 

978043 

978089 

978135 

46 

1 

8181 

8226 

8272 

8317 

8363 

8409 

8454 

8500 

8546 

8591 

46 

2 

8637 

8683 

8728 

8774 

8819 

8865 

8911 

8956 

9002 

9047 

46 

3 

9093 

9138 

9184 

9230 

9275 

9321 

9366 

9412 

9457 

9503 

46 

4 

9548 

9594 

9639 

9685 

9730 

9776 

9821 

9867 

9912 

9958 

46 

5 

980003 

980049 

980094 

980140 

980185 

980231 

980276 

980322 

980367 

980412 

45 

6 

0458 

0503 

0549 

0594 

0640 

0685 

0730 

0776 

0821 

0867 

45 

7 

0912 

0957 

1003 

1048 

1093 

1139 

1184 

1229 

1275 

1320 

45 

8 

1366 

1411 

1456 

1501 

1547 

1592 

1637 

1683 

1728 

1773 

45 

9 

1819 

1864 

1909 

1954 

2000 

2045 

2090 

2135 

2181 

2226 

45 

960 

982271 

982316 

982362 

982407 

982452 

982497 

982543 

982588 

982633 

982678 

45 

1 

2723 

2769 

2814 

2859 

2904 

2949 

2994 

3040 

3085 

3130 

45 

2 

3175 

3220 

3265 

3310 

3356 

3401 

3446 

3491 

3536 

3581 

45 

3 

3626 

3671 

3716 

3762 

3807 

3852 

3897 

3942 

3987 

4032 

45 

4 

4077 

4122 

4167 

4212 

4257 

4302 

4347 

4392 

4437 

4482 

45 

5 

4527 

4572 

4617 

4662 

4707 

4752 

4797 

4842 

4887 

4932 

45 

6 

4977 

5022 

5067 

5112 

5157 

5202 

5247 

5292 

5337 

5382 

45 

7 

5426 

5471 

5516 

5561 

5606 

5651 

5696 

5741 

5786 

5830 

45 

8 

5875 

5920 

5965 

6010 

6055 

6100 

6144 

6189 

6234 

6279 

45 

9 

6324 

6369 

6413 

6458 

6503 

6548 

6593 

6637 

6682 

6727 

45 

970 

986772 

986817 

986861 

986906 

986951 

986996 

987040 

987085 

987130 

987175 

45 

1 

7219 

7264 

7309 

7353 

7398 

7443 

7488 

7532 

7577 

7622 

45 

2 

7666 

7711 

7756 

7800 

7845 

7890 

7934 

7979 

8024 

8068 

45 

3 

8113 

8157 

8202 

8247 

8291 

8336 

8381 

8425 

8470 

8514 

45 

4 

8559 

8604 

8648 

8693 

8737 

8782 

8826 

8871 

8916 

8960 

45 

5 

9005 

9049 

9094 

9138 

9183 

9227 

9272 

9316 

9361 

9405 

45 

6 

9450 

9494 

9539 

9583 

9628 

9672 

9717 

9761 

9806 

9850 

44 

7 

9895 

9939 

9983 

990028 

990072 

990117 

990161 

990206 

990250 

990294 

44 

8 

990339 

990383 

990428 

0472 

0516 

0561 

0605 

0650 

0694 

0738 

44 

9 

0783 

0827 

0871 

0916 

0960 

1004 

1049 

1093 

1137 

1182 

44 

980 

991226 

991270 

991315 

991359 

991403 

991448 

991492 

991536 

991580 

991625 

44 

1 

1669 

1713 

1758 

1802 

1846 

1890 

1935 

1979 

2023 

2067 

44 

2 

2111 

2156 

2200 

2244 

2288 

2333 

2377 

2421 

2465 

2509 

44 

3 

2554 

2598 

2642 

2686 

2730 

2774 

2819 

2863 

2907 

2951 

44 

4 

2995 

3039 

3083 

3127 

3172 

3216 

3260 

3304 

3348 

3392 

44 

5 

3436 

3480 

3524 

3568 

3613 

3657 

3701 

3745 

3789 

3833 

44 

6 

3877 

3921 

3965 

4009 

4053 

4097 

4141 

4185 

4229 

4273 

44 

7 

4317 

4361 

4405 

4449 

4493 

4537 

4581 

4625 

4669 

4713 

44 

8 

4757 

4801 

4845 

4889 

4933 

4977 

5021 

5065 

5108 

5152 

44 

9 

5196 

5240 

5284 

5328 

5372 

5416 

5460 

5504 

6547 

6591 

44 

990 

995635 

995679 

995723 

995767 

995811 

995854 

995898 

995942 

995986 

996030 

44 

1 

6074 

6117 

6161 

6205 

6249 

6293 

6337 

6380 

6424 

6468 

44 

2 

6512 

6555 

6599 

6643 

6687 

6731 

6774 

6818 

6862 

6906 

44 

3 

6949 

6993 

7037 

7080 

7124 

7168 

7212 

7255 

7299 

7343 

44 

4 

7386 

7430 

7474 

7517 

7561 

7605 

7648 

7692 

7736 

7779 

44 

.5 

7823 

7867 

7910 

7954 

7998 

8041 

8085 

8129 

8172 

8216 

44 

6 

8259 

8303 

8347 

8390 

8434 

8477 

8521 

8564 

8608 

8652 

44 

7 

8695 

8739 

8782 

8826 

8869 

8913 

8956 

9000 

9043 

9087 

44 

8 

9131 

9174 

9218 

9261 

9305 

9348 

9392 

9435 

9479 

9522 

44 

9 

9565 

9609 

9652 

9696 

9739 

9783 

9826 

9870 

9913 

9957 

43 

No. 

0 

1 

2 

3 

1 4 

1 5 

1 6 

7 

8 

1 9 

Diff. 
































































TABLE II 


NATURAL SINES AND COSINES. 




17 




18 TABLE II. NATURAL SINES AND COSINES. 



0° 

1 ° 

2° 

3 

o 

4 

o 


M. 

Sine. 

Cos. 

Sine. 

| Cos. 

Sine. 

Cos. 

Sine. 

Cos. 

Sine. 

Cos. 

M. 

0 

.00000 

One. 

.01745 

.99985 

.03490 

.99939 

.05234 

.99863 

.06976 

.99756 

60 

1 

.00029 

One. 

.01774 

.99984 

.03519 

.99938 

.05263 

.99861 

.07005 

.99754 

59 

2 

.00058 

One. 

.01803 

,.99984 

.03548 

.99937 

.05292 

.99860 

.07034 

.99752 

58 

3 

.00087 

One. 

.01832 

.99983 

1.03577 

.99936 

.05321 

.99858 

.07063 

.99750 

57 

4 

.00116 

One. 

.01862 

.99983 

.03606 

.99935 

.05350 

.99857 

.07092 

.99748 

56 

5 

.00145 

One. 

.01891 

.999X2 

.03635 

.99934 

.05379 

.99855 

.07121 

.99746 

55 

6 

.00175 

One. 

.01920 

.99982 

.03664 

.99933 

.05408 

.99854 

.07150 

.99744 

54 

7 

.00204 

One. 

.04949 

.99981 

.03693 

.99932 

.05437 

.99852 

.07179 

.99742 

53 

8 

.00233 

One. 

.01978 

.99980 

.03723 

.99931 

.05466 

.99851 

.07208 

.99740 

52 

9 

.00262 

^One. 

.02007 

.99980 

.03752 

.99930 

.05495 

.99849 

.07237 

.99738 

51 

10 

.00291 

One. 

.02036 

.99979 

.03781 

.99929 

.05524 

.99847 

.07266 

.99736 

50 

11 

.00320 

.99999 

.02065 

.99979 

.03810 

.99927 

.05553 

.99846 

.07295 

.99734 

49 

12 

.00319 

.99999 

.02094 

.99978 

.03839 

.99926 

.05582 

.99844 

.07324 

.99731 

48 

13 

.00378 

.99999 

.02123 

.99977 

.03868 

.99925 

.05611 

.99842 

.07353 

.99729 

47 

14 

.00407 

.99999 

.02152 

.99977 

.03897 

.99924 

.05640 

.99841 

.07382 

.99727 

46 

15 

.00436 

.99999 

.02181 

.99976 

.03926 

.99923 

.05669 

.99839 

.07411 

.99725 

45 

16 

.00465 

.99999 

.02211 

.99976 

.03955 

.99922 

.05698 

.99838 

.07440 

.99723 

44 

17 

.00195 

.99999 

*02240 

.99975 

.03984 

.99921 

.05727 

.99836 

.07469 

.99721 

43 

18 

.00524 

.99999 

.02269 

.99974 

.04013 

.99919 

.05756 

.99834 

.07498 

.99719 

42 

19 

.00553 

.99998 

.02298 

.99974 

.04042 

.99918 

.05785 

.99833 

.07527 

.99716 

41 

20 

.00582 

.99998 

.02327 

.99973 

.04071 

.99917 

.05814 

.99831 

.07556 

.99714 

40 

21 

.00611 

.99998 

.02356 

.99972 

.04100 

.99916 

.05844 

.99829 

.07585 

.99712 

39 

22 

.00640 

.99998 

.02385 

.99972 

.04129 

.99915 

.05873 

.99827 

.07614 

.99710 

38 

23 

.00669 

.99998 

.02414 

.999/1 

.04159 

.99913 

.05902 

.99826 

.07643 

.99708 

37 

24 

.00698 

.99998 

.0244.3 

.99970 

.04188 

.99912 

.05931 

.99824 

.07672 

.99705 

36 

25 

.00727 

.99997 

.02472 

.99969 

.04217 

.99911 

.05960 

.99822 

.07701 

.99703 

35 

26 

.00756 

.99997 

.02501 

.99969 

.04246 

.99910 

.05989 

.99821 

.07730 

.99701 

34 

27 

.00785 

.99997 

.02530 

.99968 

.04275 

.99909 

.06018 

.99819 

.07759 

.99699 

33 

28 

.00814 

.99997 

.02560 

.99967 

.04304 

.99907 

.06047 

.99817 

.07788 

.99696 

32 

29 

.00841 

.99996 

.02589 

.99966 

.04333 

.99906 

.06076 

.99815 

.07817 

.99694 

31 

30 

.00873 

.99996 

.02618 

.99966 

.04362 

.99905 

.06105 

.99813 

.07846 

.99692 

30 

31 

.00902 

.99996 

.02647 

.99965 

.04391 

.99904 

.06134 

.99812 

.07875 

.99689 

29 

32 

.00931 

.99996 

.02676 

.99964 

.04420 

.99902 

.06163 

.99810 

.07904 

.99687 

28 

33 

.00960 

.99995 

.02705 

.99963 

.04449 

.99901 

.06192 

.99808 

.07933 

.99685 

27 

31 

.00989 

.99995 

.02734 

.99963 

.04478 

.99900 

.06221 

.99806 

.07962 

.99683 

26 

35 

.01018 

.99995 

.02763 

.99962 

.04507 

.99898 

.06250 

.99804 

.07991 

.99680 

25 

36 

.01047 

.99995 

.02792 

.99961 

.04536 

.99897 

.06279 

.99803 

.08020 

.99678 

24 

37 

.01076 

.99994 

.02821 

.99960 

.04565 

. 99896 

.06308 

.99801 

.08049 

.99676 

23 

38 

.01105 

.99994 

.02850 

.99959 

.04594 

.99894 

.06337 

.99799 

.08078 

.99673 

22 

39 

.01131 

.99994 

.02879 

.99959 

.04623 

.99893 

.06366 

.99797 

.08107 

.99671 

21 

40 

.01164 

.99993 

.02908 

.99958 

.04653 

.99892 

.06395 

.99795 

.08136 

.99668 

20 

41 

.01193 

.99993 

.02938 

.99957 

.04682 

.99890 

.06424 

.99793 

.08165 

.99666 

19 

42 

.01222 

.99993 

.02967 

. 99956 

.04711 

.99889 

.06453 

.99792 

.08194 

.99664 

18 

43 

.01251 

.99992 

.02996 

. 99955 

.04740 

.998X8 

.06482 

.99790 

.08223 

.99661 

17 

41 

.01280 

.99992 

.03025 

.99954 

.■04769 

.99886 

.06511 

99788 

.08252 

.99659 

16 

45 

.01309 

.99991 

.03054 

.99953 

.04798 

.99885 

.06540 

.99786 

.08281 

.99657 

15 

46 

.01338 

.99991 

.03083 

.99952 

.04827 

.99883 

.06569 

.99784 

.08310 

.99654 

14 

47 

.01367 

.99991 

.03112 

.99952 

.04856 

.99882 

.06598 

.99782 

.08339 

.99652 

13 

48 

.01396 

.99990 

.03141 

.99951 

.04885 

.99881 

.06627 

.99780 

.08368 

.99649 

12 

49 

.01425 

.,99990 

.03170 

.99950 

.04914 

.99879 

.06656 

.99778 

.08397 

.99647 

11 

50 

.01454 

.99989 

.03199 

.99949 

.04943 

.99878 

.06685 

.99776 

.08426 

.99644 

10 

51 

.01483 

.99989 

.03228 

.99948 

.04972 

.99876 

.06714 

.99774 

.08455 

.99642 

9 

52 

.01513 

.99989 

.03257 

.99947 

.05001 

.99875 

.06743 

.99772 

.08484 

.99639 

8 

53 

.01542 

.99988 

.03286 

.99946 

.05030 

.99873 

.06773 

.99770 

.08513 

.99637 

7 

51 

.01571 

.99988 

.03316 

.99945 

.05059 

.99872 

.06802 

.99768 

.08542 

.99635 

6 

55 

.01600 

.99987 

.03345 

.99944 

.05088 

.99870 

.06831 

.99766 

.08571 

.99632 

5 

56 

.01629 

.99987 

.03374 

.9991:: 

.05117 

.99869 

.06860 

.99764 

.08600 

.99630 

4 

57 

.01658 

.99986 

.03403 

.99942 

.05146 

.99867 

.06889 

.99762 

.08629 

.99627 

3 

58 

01687 

.99986 

.03432 

.99941 

.05175 

.99866 

.06918 

.99760 

.08658 

.99625 

2 

59 

.01716 

.99985 

.03461 

.99940 

.05205 

.99864 

.06947 

.99758 

.08687 

.99622 

1 

60 

.01745 

.99985 

.03490 

.99939 

.05234 

.99863 

.06976 

.99756 

.08716 

.99619 

0 

M. 

Cos. 

Sine. 

Cos. 

Sine. 

Cos. 

Sine. 

Cos. 

Sine. 

Cos. 

Sine. 

M. 


89 

88° 

87° 

86° 

85° 






















































































TABLE II. NATURAL SINES AND COSINES. 19 



5° 

6 

o 

7 

o 

8 


9 



M. 

Sine. 

Cos. 

Sine. 

Cos. 

Sine. 

Cos. 

Sine. 

Cos. 

Sine. 

Cos. 

M. 

0 

.08716 

.99619 

.10453 

.99452 

.12187 

.99255 

.13917 

99027 

15643 

.98769 

60 

1 

.08745 

.99617 

.10482 

.99449 

.12216 

.99251 

.13946 

.99023 

.15672 

.98764 

59 

2 

.08774 

.99614 

.10511 

.99446 

.12245 

.99248 

.13975 

.99019 

.15701 

.98760 

58 

3 

.08803 

.99612 

.10540 

.99443 

.12274 

.99244 

.14004 

.99015 

.15730 

.98755 

57 

4 

.08831 

.99609 

.10569 

.99440 

.12302 

.99240 

.14033 

.99011 

.15758 

.98751 

56 

5 

.08860 

.99607 

.10597 

.99437 

.12331 

.99237 

.14061 

.99006 

.15787 

.98746 

55 

G 

.08889 

.99604 

.10626 

.99434 

.12360 

.99233 

.14090 

.99002 

.15816 

.98741 

54 

7 

.08918 

.99602 

.10655 

.99431 

.12389 

.99230 

.14119 

.98998 

.15845 

.98737 

53 

8 

.08947 

.99599 

.10684 

.99428 

.12418 

.99226 

.14148 

.98994 

.15873 

.98732 

52 

9 

.08976 

.99596 

.10713 

.99424 

.12447 

.99222 

.14177 

.98990 

.15902 

.98728 

51 

10 

.09005 

.99594 

.10742 

.99421 

.12476 

.99219 

.14205 

.98986 

.15931 

.98723 

50 

11 

.09034 

.99591 

.10771 

.99418 

.12504 

.99215 

.14234 

.98982 

.15959 

.98718 

49 

12 

.09063 

.99588 

.10800 

.99415 

.12533 

.99211 

.14263 

.98978 

.15988 

.98714 

48 

13 

.09092 

.99586 

.10829 

.99412 

. 12562 

.99208 

.14292 

.98973 

.16017 

.98709 

47 

14 

.09121 

.99583 

.10858 

.99409 

.12591 

.99204 

.14320 

.98969 

.16046 

.98704 

46 

15 

.09150 

.99580 

.10887 

.99406 

.12620 

.99200 

.14349 

.98965 

.16074 

.98700 

45 

16 

.09179 

.99578 

.10916 

.99402 

.12649 

.99197 

.14378 

.98961 

.16103 

.98695 

44 

17 

.09208 

.99575 

.10945 

.99399 

.12678 

.99193 

.14407 

.98957 

.16132 

.98690 

43 

18 

.09237 

.99572 

.10973 

.99396 

.12706 

.99189 

.14436 

.98953 

.16160 

.98686 

42 

19 

.09266 

.99570 

.11002 

.99393 

.12735 

.99186 

.14464 

.98948 

.16189 

.98681 

41 

20 

.09295 

.99567 

.11031 

.99390 

.12764 

.99182 

.14493 

.98944 

.16218 

.98676 

40 

21 

.09324 

.99564 

.11060 

.99386 

.12793 

.99178 

.14522 

.98940 

.16246 

.98671 

39 

22 

.09353 

.99562 

.11089 

.99383 

.12822 

.99175 

.14551 

.98936 

.16275 

.98667 

38 

23 

.09382 

.99559 

.11118 

.99380 

.12851 

.99171 

.14580 

.98931 

.16304 

.98662 

37 

21 

.09411 

.99556 

.11147 

.99377 

.12880 

.99167 

.14608 

.98927 

.16333 

.98657 

36 

25 

.09440 

.99553 

.11176 

.99374 

.12908 

.99163 

.14637 

.98923 

.16361 

.98652 

35 

26 

.09469 

.99551 

.11205 

.99370 

.12937 

.99160 

.14666 

.98919 

.16390 

.98648 

34 

27 

.09498 

.99548 

.11234 

.99367 

.12966 

.99156 

.14695 

.98914 

.16419 

.98643 

33 

28 

.09527 

.99545 

.11263 

.99364 

.12995 

.99152 

.14723 

.98910 

.16447 

.98638 

32 

29 

.09556 

.99542 

.11291 

.99360 

. 13024 

.99148 

.14752 

.98906 

.16476 

.98633 

31 

30 

.09585 

.99540 

.11320 

.99357 

. 13053 

.99144 

.14781 

.98902 

.16505 

.98629 

30 

31 

.09614 

.99537 

.11349 

.99354 

. 13081 

.99141 

.14810 

.98897 

.16533 

.98624 

29 

32 

.09642 

.99534 

.11378 

.99351 

.13110 

.99137 

.14838 

.98893 

.16562 

.98619 

28 

33 

.09671 

.99531 

.11407 

.99347 

.13139 

.99133 

.14867 

.98889 

.16591 

.98614 

27 

34 

.09700 

.99528 

.11436 

.99344 

.13168 

. 99129 

.14896 

.98884 

.16620 

.98609 

26 

35 

.09729 

.99526 

.11465 

.99341 

.13197 

. 99125 

.14925 

.98880 

.16648 

.98604 

25 

36 

.09758 

.99523 

.11494 

.99337 

.13226 

. 99122 

.14954 

.98876 

.16677 

.98600 

24 

37 

.09787 

.99520 

.11523 

.99334 

.13254 

.99118 

.14982 

.98871 

.16706 

.98595 

23 

38 

.09816 

.99517 

.11552 

.99331 

.13283 

.99114 

.15011 

.98867 

.16734 

.98590 

22 

39 

.09845 

.99514 

.11580 

.99327 

.13312 

. 99110 

.15040 

.98863 

.16763 

.98585 

21 

40 

.09874 

.99511 

.11609 

.99324 

. 13341 

. 99106 

.15069 

.98858 

.16792 

.98580 

20 

41 

.09903 

.99508 

.11638 

.99320 

.13370 

. 99102 

.15097 

.98854 

.16820 

.98575 

19 

42 

.09932 

.99506 

.11667 

.99317 

. 13399 

. 99098 

.15126 

.98849 

.16849 

.98570 

18 

43 

.09961 

.99503 

.11696 

.99314 

. 13427 

. 99094 

.15155 

.98845 

.16878 

.98565 

17 

44 

.09990 

.99500 

.11725 

.99310 

. 13456 

.99091 

.15184 

.98841 

.16906 

98561 

16 

45 

.10019 

.99497 

.11754 

.99307 

. 13485 

.99087 

.15212 

.98836 

.16935 

1.98556 

15 

46 

.10048 

.99494 

.11783 

.99303 

. 13514 

.99083 

1 .15241 

.98832 

.16964 

.98551 

14 

47 

.10077 

.99491 

.11812 

.99300 

. 13543 

.99079 

.15270 

.98827 

.16992 

.98546 

13 

48 

.10106 

.99488 

.11840 

.99297 

. 13572 

. 99075 

.15299 

.98823 

.17021 

.98541 

12 

49 

.10135 

.99485 

.11869 

.99293 

. 13600 

.99071 

.15327 

.98818 

.17050 

.98536 

11 

50 

.10164 

.99482 

.11898 

.99290 

. 13629 

.99067 

.15356 

.98814 

.17078 

.98531 

10 

51 

.10192 

.99479 

.11927 

.99286 

. 13658 

.99063 

.15385 

.98809 

.17107 

.98526 

9 

52 

.10221 

.99476 

.11956 

.99283 

|. 13687 

.99059 

T5414 

.98805 

.17136 

.98521 

8 

53 

.10250 

.99473 

.11985 

.99279 

).13716 

.99055 

.15442 

.98800 

.17164 

.98516 

7 

54 

.10279 

.99470 

.12014 

.99276 

!.13744 

.99051 

.15471 

.98796 

.17193 

.98511 

6 

55 

.10308 

.99467 

.12043 

.99272 

.13773 

.99047 

.15500 

.98791 

.17222 

.98506 

5 

56 

.10337 

.99464 

.12071 

.99269 

.13802 

.99043 

.15529 

.98787 

.17250 

.98501 

4 

57 

.10366 

.99461 

.12100 

.99265 

.13831 

.99039 

.15557 

.98782 

.17279 

.98496 

3 

58 

.10395 

.99458 

.12129 

.99262 

.13860 

.99035 

.15586 

.98778 

.17308 

.98491 

2 

59 

.10424 

.99455 

.12158 

.99258 

.13889 

.99031 

.15615 

.98773 

.17336 

.98486 

1 

60 

.10453 

.99452 

.12187 

.99255 

.13917 

.99027 

.15643 

.98769 

.17365 

.98481 

0 

M. 

Cos. 

1 Sine. 

Cos. 

Sine. 

Cos, 

Sine. 

Cos. 

Sine. 

Cos. 

Sine. 

M. 


84° 

83° 

o 

00 

1 

81° 

80 s 















































































20 TABLE II. NATURAL SINES AND COSINES. 



10° 

11° 

12° 

13° 

14° 


M. 

Sine. 

Cos. 

Sine. 

Cos. 

Sine. 

Cos. 

Sine. I 

Cos. 

Sine. 

Cos. 

M. 

0 

.17365 

.98481 

.19081 

.98163 

.20791 

.97815 

.22495 

.97437 

.24192 

.97030 

60 

1 

.17393 

.98476 

.19109 

.98157 

.20820 

.97809 

.22523 

.97430 

.24220 

.97023 

59 

2 

.17422 

.98471 

.19138 

.98152 

.20848 

.97803 

.22552 

.97424 

.24249 

.97015 

58 

3 

.17451 

.98466 

.19167 

.98146 

.20877 

.97797 

.22580 

.97417 

.24277 

.97008 

57 

4 

.17479 

.98461 

.19195 

.98140 

.20905 

.97791 

.22608 

.97411 

.24305 

.97001 

56 

5 

.17508 

.98455 

.19224 

.98135 

.20933 

.97784 

.22637 

.97404 

.24333 

.96994 

55 

6 

.17537 

.98450 

.19252 

.98129 

.20962 

.97778 

.22665 

.97398 

.24362 

.96987 

54 

7 

.17505 

.98445 

.19281 

.98124 

.20990 

.97772 

.22693 

.97391 

.24390 

.96980 

53 

8 

.17594 

.98440 

.19309 

.98118 

.21019 

.97766 

.22722 

.97384 

.24418 

.96973 

52 

9 

.17G23 

.98435 

.19338 

.98112 

.21047 

.97760 

.22750 

.97378 

.24446 

.96966 

51 

10 

.17G51 

.98430 

.19366 

.98107 

.21076 

.97754 

.22778 

.97371 

.24474 

.96959 

50 

11 

.17G80 

.98425 

.19395 

.98101 

.21101 

.97748 

.22807 

.97365 

.24503 

.96952 

49 

12 

.17708 

.98420 

.19423 

.98096 

.21132 

.97742 

.22835 

.97358 

.24531 

.96945 

48 

13 

.17737 

.98414 

.19452 

.98090 

.21161 

.97735 

.22863 

.97351 

.24559 

.96937 

47 

14 

.177G6 

.98409 

.19481 

.98084 

.21139 

.97729 

.22892 

.97345 

.24587 

.96930 

46 

15 

.17794 

.98404 

.19509 

.98079 

.21218 

.97723 

.22920 

.97338 

.24615 

.96923 

45 

16 

.17823 

.98399 

.19538 

.98073 

.21246 

.97717 

.22948 

.97331 

.24644 

.96916 

44 

17 

.17852 

.98394 

.19566 

.98067 

.21275 

.97711 

.22977 

.97325 

.24672 

.96909 

43 

18 

.17880 

.98389 

.19595 

.98061 

.21308 

.97705 

.23005 

.97318 

.24700 

.96902 

42 

19 

.17909 

.98383 

.19623 

.98056 

.21331 

.97698 

.23033 

.97311 

.24728 

.96894 

41 

20 

.17937 

.98378 

.19652 

.98050 

.21360 

.97692 

.23062 

.97304 

.24756 

.96887 

40 

21 

.1796G 

.98373 

.19680 

.98044 

.21388 

.97686 

.23090 

.97298 

.24784 

.96880 

39 

22 

.17995 

.98368 

.19709 

.98039 

.21417 

.97680 

.23118 

.97291 

.24813 

.96873 

38 

23 

.18023 

.98362 

.19737 

.98033 

.21445 

.97673 

.23146 

.97284 

.24841 

.96866 

37 

24 

.18052 

.98357 

.19766 

.98027 

.21474 

.97667 

.23175 

.97278 

.24869 

.96858 

36 

25 

.18081 

.98352 

.19794 

.98021 

.21502 

.97661 

.23203 

.97271 

.24897 

.96851 

35 

26 

.18109 

.98347 

.19823 

.98016 

.21530 

.97655 

.23231 

.97264 

.24925 

.96844 

34 

27 

.18138 

.98341 

.19851 

.98010 

.21559 

.97648 

.23260 

.97257 

.24953 

.96837 

33 

28 

.18166 

.98336 

.19880 

.98004 

.21587 

.97642 

.23288 

.97251 

.24982 

.96829 

32 

29 

.18195 

.98331 

.19908 

.97998 

.21616 

.97636 

.23316 

.97244 

.25010 

.96822 

31 

30 

.18224 

.98325 

.19937 

.97992 

.21644 

.97630 

.23345 

.97237 

.25038 

.96815 

30 

31 

.18252 

.98320 

.19965 

.97987 

.21672 

.97623 

.23373 

.97230 

.25066 

.96807 

29 

32 

.18281 

.98315 

.19994 

.97981 

.21701 

.97617 

.23401 

.97223 

.25094 

.96800 

28 

33 

.18309 

.98310 

.20022 

.97975 

.21729 

.97611 

.23429 

.97217 

.25122 

.96793 

27 

34 

.18338 

.98304 

.20051 

.97969 

.21758 

.97604 

.23458 

.97210 

.25151 

.96786 

26 

35 

.18367 

.98299 

.20079 

.97963 

.21786 

.97598 

.23486 

.97203 

.25179 

.96778 

25 

36 

.18395 

.98294 

.20108 

.97958 

.21814 

.97592 

.23514 

.97196 

.25207 

.96771 

24 

37 

.18424 

.98288 

.20136 

.97952 

.21843 

.97585 

.23542 

.97189 

.25235 

.96764 

23 

38 

.18452 

.98283 

.20165 

.97946 

.21871 

.97579 

.23571 

.97182 

.25263 

.96756 

22 

39 

.18481 

.98277 

.20193 

.97940 

.21899 

.97573 

.23599 

.97176 

.25291 

.96749 

21 

40 

.18509 

.98272 

.20222 

.97934 

.21928 

.97566 

.23627 

.97169 

.25320 

.96742 

20 

41 

.18538 

.98267 

.20250 

.97928 

.21956 

.97560 

.23656 

.97162 

.25348 

.96734 

19 

42 

.18567 

.98261 

.20279 

.97922 

.21985 

.97553 

.23684 

.97155 

.25376 

.96727 

18 

43 

.18595 

.98256 

.20307 

.97916 

.22013 

.97547 

.23712 

.97148 

.25404 

.96719 

17 

44 

.18624 

.98250 

.20336 

.97910 

.22041 

.97541 

.23740 

.97141 

.25432 

.96712 

16 

45 

.18652 

.98245 

.20364 

.97905 

.22070 

.97534 

.23769 

.97134 

.25460 

.96705 

15 

46 

.18681 

.98240 

.20393 

.97899 

.22098 

.97528 

.23797 

.97127 

.25488 

.96697 

14 

47 

.18710 

.98234 

.20421 

.97893 

.22126 

.97521 

.23825 

.97120 

.25516 

.96690 

13 

48 

.18738 

.98229 

.20450 

.97887 

.22155 

.97515 

.23853 

.97113 

.25545 

.96682 

12 

49 

.18767 

.98223 

.20478 

.97881 

.22183 

.97508 

. 23882 

.97106 

.25573 

.96675 

11 

50 

.18795 

.98218 

.20507 

.97875 

.22212 

.97502 

.23910 

.97100 

.25601 

.96667 

10 

51 

.18824 

.98212 

.20535 

.97869 

.22240 

.97496 

.23938 

.97093 

.25629 

.96660 

9 

52 

.18852 

.98207 

.20563 

.97863 

.22268 

.97489 

.23966 

.97086 

.25657 

.96653 

8 

53 

.18881 

.98201 

.20592 

.97857 

.22297 

.97483 

.23995 

.97079 

.25685 

.96645 

7 

54 

.18910 

.98196 

.20620 

.97851 

.22325 

.97476 

.24023 

.97072 

.25713 

.96638 

6 

55 

.18938 

.98190 

.20649 

.97845 

.22353 

.97470 

.24051 

.97065 

.25741 

.96630 

5 

56 

.18967 

.98185 

.20677 

.97839 

.22382 

.97463 

i. 24079 

.97058 

.25769 

96623 

4 

57 

.18995 

.98179 

.20706 

.97833 

.22410 

.97457 

.24108 

.97051 

.25798 

.96615 

3 

53 

.19024 

.98174 

.20734 

.97827 

.22438 

.97450 

.24136 

.97044 

.25826 

.96608 

2 

59 

.19052 

.98168 

.20763 

.97821 

.22467 

.97444 

.24164 

.97037 

.25854 

.96600 

1 

CO 

.19081 

.98163 

.20791 

.97815 

.22495 

.97437 

.24192 

.97030 

.25882 

.96593 

0 

M. 

Cos. 

Sine. 

Cos. 

Sine. 

Cos. 

Sine. 

COS. 

Sine. 

Cos. 

Sine. 

M. 


79° 

78° 

77° 

76° 

75° 


































































TABLE II. NATURAL SINES ANL> COSINES. 21 



K 


I«° 

17° 

18 

o 

19 

8 


M. 

Sine. 

Cos. 

Sine. 

Cos. 

Sine. 

Cos. 

Sine. 

Cos. 

Sine. 

Cos. 

M. 

0 

.25882 

.96593 

.27564 

.96126 

.29237 

.95630 

.30902 

.95106 

.32557 

.94552 

60 

1 

.25910 

.96585 

.27592 

.96118 

.2! 1265 

.95622 

.30929 

.95097 

.32584 

.94542 

59 

2 

.25938 

.96578 

.27620 

.96110 

.29293 

.95613 

.30957 

.95088 

.32612 

.94533 

68 

3 

.25966 

.96570 

.27648 

.96102 

.29321 

.95605 

.30985 

.95079 

.32639 

.94523 

57 

4 

.25994 

.96562 

.27676 

.91 >094 

.29348 

.95596 

.31012 

.95070 

.32667 

.94514 

56 

5 

.26022 

.96555 

.27704 

.96086 

.29376 

.95588 

.31040 

.95061 

.32694 

.94504 

55 

G 

.26050 

.96547 

.27731 

.96078 

.29404 

.95579 

.31068 

.95052 

.32722 

.94495 

54 

7 

.26079 

.96540 

.27759 

.96070 

.29432 

.95571 

.31095 

.95043 

.32749 

.94485 

53 

8 

.26107 

.96532 

.27787 

.96062 

.29460 

.95562 

.31123 

.95033 

.32777 

.94476 

52 

9 

.26135 

.96524 

.27815 

.96054 

.29487 

.95554 

.31151 

.95024 

.32804 

.94466 

51 

10 

.26163 

.96517 

.27843 

.96046 

.29515 

.95545 

.31178 

.95015 

.32832 

.94457 

50 

11 

.26191 

.96509 

.27871 

.96037 

.29.543 

.95536 

.31206 

.95006 

.32859 

.94447 

49 

12 

.26219 

.96502 

.27899 

.96029 

.29571 

.95528 

.31233 

.94997 

.32887 

.94438 

48 

13 

.26247 

.96494 

.27927 

.96021 

.29599 

.95519 

.31261 

.94988 

.32914 

.94428 

47 

14 

.26275 

.96486 

.27955 

.96013 

.29626 

.95511 

.31289 

.94979 

.32942 

.94418 

46 

15 

.26303 

.96479 

.27983 

.96005 

.29654 

.95502 

.31316 

.94970 

.32969 

.94400 

45 

16 

.26331 

.96471 

.28011 

.95997 

.29682 

.95493 

.31344 

.94961 

.32997 

.94390 

44 

17 

.26359 

.96463 

.28039 

.95989 

.29710 

.95485 

.31372 

.94952 

.33024 

.94390 

43 

18 

.26387 

.96456 

.28067 

.95981 

.29737 

.95476 

.31399 

.94943 

.33051 

.94380 

42 

19 

.26415 

.96448 

.28095 

.95972 

.29765 

.95467 

.31427 

.94933 

.33079 

.94370 

41 

20 

.26443 

.96440 

.28123 

.95964 

.29793 

.95459 

.31454 

.94924 

.33106 

.94361 

40 

21 

.26471 

.96433 

.28150 

.95956 

.29821 

.95450 

.31482 

.94915 

.33134 

.94351 

39. 

22 

.26500 

.96425 

.28178 

.95948 

.29849 

.95441 

.31510 

.94906 

.33161 

.94342 

38 

23 

.26528 

.96417 

.28206 

.95940 

.29876 

.95433 

.31537 

.94897 

.33189 

.94332 

37 

24 

.26556 

.96410 

.28234 

.95931 

.29904 

.95424 

.31565 

.94888 

.33216 

.94322 

36 

25 

.26584 

.96402 

.28262 

.95923 

.29932 

.95415 

.31593 

.94878 

.33244 

.94313 

35 

26 

.26612 

.96394 

.28290 

.95915 

.29960 

.95407 

.31620 

.94869 

.33271 

.94303 

34 

27 

.26640 

.96386 

.28318 

.95907 

.29987 

.95398 

.31648 

.948G0 

.33298 

.94293 

33 

28 

.2666S 

.96379 

.28346 

.95898 

.30015 

.95389 

.31675 

.94851 

.33326 

.94284 

32 

29 

.26696 

.96371 

.28374 

.95890 

.30043 

.95380 

.31703 

.94842 

.33353 

.94274 

31 

30 

.26724 

.96363 

.28402 

. 95882 

.30071 

.95372 

.31730 

.94832 

.33381 

.94264 

30 

31 

.26752 

.96355 

.28429 

.95874 

.30098 

.95363 

.31758 

.94823 

.33408 

.94254 

29 

32 

.26780 

.96347 

.28457 

. 95865 

.30126 

.95354 

.31786 

.94814 

.33436 

.94245 

28 

33 

.26808 

.96340 

.28485 

.95857 

.30154 

.95345 

.31813 

.94805 

.33463 

.94235 

27 

34 

.26836 

.96332 

.28513 

.95849 

.30182 

.95337 

.31841 

.94795 

.33490 

.94225 

26 

35 

.26864 

.96324 

.28541 

.95841 

.30209 

.95328 

.31868 

.94786 

.33518 

.94215 

25 

36 

.26892 

.96316 

.28569 

.95832 

.30237 

.95319 

.31896 

.94777 

.33545 

.94206 

24 

37 

.26920 

.96308 

.28597 

.95824 

.30265 

.95310 

.31923 

.94768 

.33573 

.94196 

23 

38 

.26948 

.96301 

.28625 

.95816 

.30292 

.95301 

.31951 

.94758 

.33600 

.94186 

22 

39 

.26976 

.96293 

.28652 

.95807 

.30320 

.95293 

.31979 

.94749 

.33627 

.94176 

21 

40 

.27001 

.96285 

.28680 

.95799 

.30348 

.95284 

.32006 

.94740 

.33655 

.94167 

20 

41 

.27032 

.96277 

.28708 

.95791 

.30376 

.95275 

.32034 

.94730 

.33682 

.94157 

19 

42 

.27060 

.96269 

.28736 

.95782 

.30403 

.95266 

.32061 

.94721 

.33710 

.94147 

18 

43 

.27088 

.96261 

.28764 

.95774 

.30431 

.95257 

.32089 

.94712 

.33737 

.94137 

17 

41 

.27116 

.96253 

.28792 

.95766 

.30459 

.95248 

.32116 

.94702 

.33764 

.94127 

16 

45 

.27144 

.96246 

.28820 


.30486 

.95240 

.32144 

.94693 

.33792 

.94118 

15 

46 

.27172 

.96238 

.28847 

.95749 

.30514 

.95231 

.32171 

.94684 

.33819 

.94108 

14 

47 

.27200 

.96230 

.28875 

.95740 

.30542 

.95222 

.32199 

.94674 

.33846 

.94098 

13 

48 

.27228 

.96222 

.28903 

.95732 

.30570 

.95213 

.32227 

.94665 

.33874 

.94088 

12 

49 

.27256 

.96214 

.28! «31 

.95724 

.30597 

.95204 

.32254 

.94656 

.33901 

.94078 

11 

50 

.27284 

.96206 

.28959 

.95715 

.30625 

.95195 

.32282 

.94646 

.33929 

.94068 

10 

51 

.27312 

.96198 

.28987 

.95707 

.30653 

.95186 

.32309 

.94637 

.33956 

.94058 

9 

52 

.27340 

.96190 

.29015 

.95698 

.30680 

.95177 

1.32337 

.94627 

.33983 

.94049 

8 

53 

.27368 

.96182 

.29042 

.95690 

. 30708 

f. 

y —l 

o 

Ci 

.32364 

.94618 

.34011 

.94039 

7 

54 

.27396 

.96174 

.29070 

.95681 

.30736 

.95159 

.32392 

.94609 

.34038 

.94029 

6 

55 

.27424 

.96166 

.29098 

.95673 

. 30763 

.95150 

.32419 

.94599 

.34065 

.94019 

5 

56 

.27452 

.96158 

.29126 

.95664 

.30791 

.95142 

.32447 

.94590 

.34093 

.94009 

4 

57 

.27480 

.96150 

.29154 

.95656 

.30819 

.95133 

.32474 

.94580 

.34120 

.93999 

3 

58 

.27508 

.96142 

.29182 

.95647 

.30846 

.95124 

.32502 

.94571 

.34147 

.93989 

2 

59 

.27536 

.96134 

.29209 

.95639 

.30874 

.95115 

.32529 

.94561 

.34175 

.93979 

1 

60 

.27564 

.96126 

.29237 

.95630 

.30902 

.95106 

.32557 

.94552 

.34202 

.93969 

0 

M. 

Cos. 

Sine. 

Cos. 

Sine. 

Cos. 

Sine. 

Cos. 

Sine. 

Cos. 

Sine. 

M. 


74° 

7 

3 

72° 

4 ** 

7 

1° 

70° 



















































































22 TABLE II. NATURAL SINES AND COSINES. 



20° 

21° 

22° 

23° 

24° 


M. 

Sine. 

Cos. 

Sine. 

Cos. 

S ine. 

Cos. 

Sine. 

Cos. 

Sine. 

Cos. 

M. 

0 

.34202 

.93969 

.35837 

.93358 

.37461 

.92718 

.39073 

.92050 

.40674 

.91355 

60 

1 

.34229 

.93959 

.35864 

.93348 

.37488 

.92707 

.39100 

.92039 

.40700, 

.91343 

59 

2 

.34257 

.93949 

.35891 

.93337 

.37515 

.92697 

.39127 

.92028 

.40727 

.91331 

58 

3 

.34284 

.93939 

.35918 

.93327 

.37542 

.92686 

.39153 

.92016 

.40753 

.91319 

57 

4 

.34311 

.93929 

.35945 

.93316 

.37569 

.92675 

.39180 

.92005 

.40780 

.91307 

50 

5 

.34339 

.93919 

.35973 

.93306 

.37595 

.92664 

.39207 

.91994 

.40806 

.91295 

55 

6 

.34366 

.93909 

.36000 

.93295 

.37622 

.92653 

.39234 

.91982 

.40833 

.91283 

54 

7 

.34393 

.93899 

.36027 

.93285 

.37649 

.92642 

.39200 

.91971 

.40860 

.91272 

53 

8 

.34421 

.93889 

.36054 

.93274 

.37676 

.92631 

.39287 

.91959 

.40886 

.91260 

52 

9 

.34448 

.93879 

.36081 

.93264 

.37703 

.92620 

.39314 

.91948 

.40913 

.91248 

51 

10 

.34475 

.93869 

.36108 

.93253 

.37730 

.92609 

.39341 

.91936 

.40939 

.91236 

50 

11 

.34503 

.93859 

.36135 

.93243 

.37757 

.92598 

.39367 

.91925 

.40966 

.91224 

49 

12 

.34530 

.93849 

.36162 

.93232 

.37784 

.92587 

.39394 

.91914 

.40992 

.91212 

48 

13 

.34557 

.93839 

.36190 

.93222 

.37811 

.92576 

.39421 

.91902 

.41019 

.91200 

47 

14 

.34584 

.93829 

.36217 

.93211 

.37838 

.92565 

.39448 

.91891 

.41045 

.91188 

46 

15 

.34612 

.93819 

.36244 

.93201 

.37865 

.92554 

.39474 

.91879 

.41072 

.91176 

45 

16 

.34639 

.93809 

.36271 

.93190 

.37892 

.92543 

.39501 

.91868 

.41098 

.91164 

44 

17 

.34666 

.93799 

.36298 

.93180 

.37919 

.92532 

.39528 

.91856 

.41125 

.91152 

43 

18 

.34694 

.93789 

.36325 

.93169 

.37946 

.92521 

.39555 

.91845 

.41151 

.91140 

42 

19 

.34721 

.93779 

.36352 

.93159 

.37973 

.92510 

.39581 

.91833 

.41178 

.91128 

41 

20 

.34748 

.93769 

.36379 

.93148 

.37999 

.92499 

.39608 

.91822 

.41204 

.91116 

40 

21 

.34775 

.93759 

.36406 

.93137 

.38026 

.92488 

.39635 

.91810 

.41231 

.91104 

39 

22 

.34803 

.93748 

.36434 

.93127 

.38053 

.92477 

.39661 

.91799 

.41257 

.91092 

38 

23 

.34830 

.93738 

..".6461 

.93116 

.38080 

.92466 

.39688 

.91787 

.41284 

.91080 

37 

24 

.34857 

.93728 

.36488 

.93106 

.38107 

.92455 

.39715 

.91775 

.41310 

.91068 

30 

25 

.34884 

.93718 

.36515 

.93095 

.38134 

.92444 

.39741 

.91764 

.41337 

.91056 

35 

26 

.34912 

.93708 

.36542 

.93084 

.38161 

.92432 

.39768 

.91752 

.41363 

.91044 

34 

27 

.34939 

.93698 

.36569 

.93074 

.38188 

.92421 

.39795 

.91741 

.41390 

.91032 

33 

28 

.34966 

.93688 

.36596 

.93063 

.38215 

.92410 

.39822 

.91729 

.41416 

.91020 

32 

29 

.34993 

.93677 

.36623 

.93052 

.38241 

.92399 

.39848 

.91718 

.41443 

.91008 

31 

30 

.35021 

.93667 

.36650 

.93042 

.38268 

.92388 

.39875 

.91706 

.41469 

.90996 

30 

31 

.35048 

.93657 

.36677 

.93031 

.38295 

.92377 

.39902 

.91694 

.41496 

.90984 

29 

32 

.35075 

.93647 

.36704 

.93020 

.38322 

.92366 

.39928 

.91683 

.41522 

.90972 

28 

33 

.35102 

.93637 

.36731 

.93010 

.38349 

.92355 

.39955 

.91671 

.41549 

.90960 

27 

34 

.35130 

.93626 

.36758 

.92999 

.38376 

.92343 

.39982 

.91660 

.41575 

. 1X1948 

26 

35 

.35157 

.93616 

.36785 

.92988 

.38403 

.92332 

.40008 

.91648 

.41602 

.90936 

25 

36 

.35184 

.93606 

.36812 

.92978 

.38430 

.92321 

.40035 

.91636 

.41628 

.90924 

24 

37 

.35211 

.93596 

.36839 

.92967 

.38456 

.92310 

.40062 

.91625 

.41655 

.90911 

23 

38 

.35239 

.93585 

.36867 

.92956 

.38483 

.92299 

.40088 

.91613 

.41681 

.90899 

22 

39 

.35266 

.93575 

.36894 

.92945 

.38510 

.92287 

.40115 

.91601 

.41707 

.90887 

21 

40 

.35293 

.93565 

.36921 

.92935 

.38537 

.92276 

.40141 

.91590 

.41734 

.90875 

20 

41 

.35320 

.93555 

.36948 

.92924 

.38564 

.92265 

.40168 

.91578 

.41760 

.90863 

19 

42 

.35347 

.93544 

.36975 

.92913 

.38591 

.922.54 

.40195 

.91566 

.41787 

.90851 

18 

43 

.35375 

.93534 

.37002 

.92902 

.38617 

.92243 

.40221 

.91555 

.41813 

.90839 

17 

44 

.35402 

.93524 

.37029 

.92892 

.38644 

.92231 

.40248 

.91543 

.41840 

.90826 

16 

45 

.35429 

.93514 

.37056 

.92881 

.38671 

.92220 

.40275 

.91531 

.41866 

.90814 

15 

46 

.35456 

.93503 

.37083 

.92870 

.38698 

.92209 

.40301 

.91519 

.41892 

.90802 

14 

47 

.35484 

.93493 

.37110 

.92859 

.38725 

.92198 

.40328 

.91508 

.41919 

.90790 

13 

48 

.35511 

.93483 

.37137 

.92849 

.38752 

.92186 

.40355 

.91496 

.41945 

.90778 

12 

49 

.35538 

.93472 

.37164 

.92838 

.38778 

.92175 

.40381 

.91484 

.41972 

.90760 

11 

50 

.35565 

.93462 

.37191 

,92S‘J7 

.38805 

.92164 

.40408 

.91472 

.41998 

.90753 

10 

51 

.35592 

.93452 

.37218 

.92816 

.38832 

.921^2 

.40434 

.91461 

.42024 

.90741 

9 

52 

. 356l!J 

.93441 

.37245 

.92805 

.38859 

.92141 

.40461 

.91449 

.42051 

.90729 

8 

53 

.35647 

.93431 

.37272 

.92794 

.38886 

.92130 

.40488 

.91437 

.42077 

.90717 

7 

54 

.356 I 4 

.9.3120 

.37299 

.92784 

.38912 

.92119 

.40514 

.91425 

.42104 

.90704 

6 

55 

.35701 

.93410 

.37326 

.92773 

.3X939 

.92107 

.40541 

.91414 

.42130 

.90692 

5 

56 

.35728 

.93400 

.37353 

.92762 

.38966 

.92096 

.40567 

.91402 

.42156 

.90680 

4 

67 

.35755 

.93389 

.37380 

.92751 

.38993 

.92085 

.40594 

.91390 

.42183 

.90668 

3 

58 

.35782 

.93379 

.37407 

.92740 

.39020 

.92073 

.40621 

.91378 

.42209 

.90655 

2 

59 

.35810 

.93368 

.37434 

.92729 

.3!H140 

.92062 

.40647 

.91366 

.42235 

.90043 

1 

60 

. 35wt 

.93358 

.37461 

.92718 

.39073 

.92050 

.40674 

.91355 

.42262 

.90631 

0 

M. 

Cos. 

1 Sine. 

Cos. 

Sine. 

Cos. 

Sine. 

Cos. 

Sine, 

Cos. 

Sine. 

M. 


69° 

68° 

67° 

66° 

65° 





























































TABLE II. NATURAL SINES AND COSINES. 23 



o 

5° 

26° 

2 

7° 

28° 

29° 


M. 

Sine. 

Cos. 

Sine. 

Cos. 

Sine. 

Cos. 

Sine. 

Cos. 

Sine. 

Cos. 

M. 

0 

.42262 

.90631 

.43837 

.89879 

.45399 

.89101 

.46947 

.88295 

.48481 

.87462 

60 

1 

.42288 

.90618 

.43863 

.89867 

.45425 

.89087 

.46973 

.88281 

.48506 

.87448 

59 

2 

.42315 

.90606 

.43889 

.89854 

.45451 

.89074 

.46999 

.88267 

.48532 

.87434 

58 

3 

.42341 

.90594 

.43916 

.89841 

.45477 

.89061 

.47024 

.88254 

.48557 

.87420 

57 

4 

.42367 

.90582 

.43942 

.89828 

.45503 

.89048 

.47050 

.88240 

.48583 

.87406 

56 

5 

.42394 

-.90569 

.43968 

.89816 

.45529 

.890.35 

.47076 

.88226 

.48608 

.87391 

55 

6 

.42420 

.90557 

.43994 

.89803 

.45554 

.89021 

.47101 

.88213 

.48634 

.87377 

54 

7 

.42446 

.90545 

. 44020 

.89790 

.45580 

.89008 

.47127 

.88199 

.48659 

.87363 

53 

8 

.42473 

.90532 

.44046 

.89777 

.45606 

.88995 

.47153 

.88185 

.48684 

.87349 

52 

9 

.42499 

.90520 

.44072 

.89764 

.45632 

.88981 

.47178 

.88172 

.48710 

.87335 

51 

10 

.42525 

.90507 

.44098 

.89752 

.45658 

.88968 

.47204 

.88138 

.48735 

.87321 

50 

11 

.42552 

.90495 

.44124 

.89739 

.45684 

.88955 

.47229 

.88144 

.48761 

.87306 

49 

12 

.42578 

.90483 

.44151 

.89726 

.45710 

.88942 

.47255 

.88130 

.48786 

.87292 

48 

13 

.42604 

.90470 

.44177 

.89713 

.45736 

.88928 

.47281 

.88117 

.48811 

.87278 

47 

14 

.42631 

.90458 

.44203 

.89700 

.45762 

.88915 

.47306 

.88103 

.48837 

.87264 

46 

15 

.42657 

.90446 

.44229 

.89687 

.45787 

.88902 

.47332 

.88089 

.48862 

.87250 

45 

16 

.42683 

.90433 

.44255 

.89674 

.45813 

.88888 

.47358 

.88075 

.48888 

.87235 

44 

17 

.42709 

.90421 

.44281 

.89662 

.45839 

.88875 

.47383 

.88062 

.48913 

.87221 

43 

18 

.42736 

.90408 

.44307 

.89649 

.45865 

.88862 

.47409 

.88048 

.48938 

.87207 

42 

19 

.42762 

.90396 

.44333 

.89636 

.45891 

.88848 

.47434 

.88034 

.48964 

.87193 

41 

20 

.42788 

.90383 

.44359 

.89623 

.45917 

.88835 

.47460 

.88020 

.48989 

.87178 

40 

21 

.42815 

.90371 

.44385 

.89610 

.45942 

.88822 

.47486 

.88006 

.49014 

.87164 

39 

22 

.42841 

.90358 

.44411 

.89597 

.45968 

.88808 

.47511 

.87993 

.49040 

.871.50 

38 

23 

.42867 

.90346 

.44437 

.89584 

.45994 

.88795 

.47537 

.87979 

.49065 

.87136 

37 

24 

.42894 

.90334 

.44464 

.89571 

.46020 

.88782 

.47562 

.87965 

.49090 

.87121 

36 

25 

.42920 

.90321 

.44490 

.89558 

.46046 

.88768 

.47588 

.87951 

.49116 

.87107 

35 

26 

.42946 

.90309 

.44516 

.89545 

.46072 

.88755 

.47614 

.87937 

.49141 

.87093 

34 

27 

.42972 

.90296 

.44542 

.89532 

.46097 

.88741 

.47639 

.87923 

.49166 

.87079 

33 

28 

.42999 

.90284 

.44568 

.89519 

.46123 

.88728 

.47665 

.87909 

.49192 

.87064 

32 

29 

.43025 

.90271 

.44594 

.89506 

.46149 

.88715 

.47690 

.87896 

.49217 

.87050 

31 

30 

.43051 

.90259 

.44620 

.89493 

.46175 

.88701 

.47716 

.87882 

.49242 

.87036 

30 

31 

.43077 

.90246 

.44646 

.89480 

.46201 

.88688 

.47741 

.87868 

.49268 

.87021 

29 

32 

.43104 

.90233 

.44672 

.89467 

.46226 

.88674 

.47767 

.87854 

.49293 

.87007 

28 

33 

.43130 

.90221 

.44698 

.89454 

.46252 

.88661 

.47793 

.87840 

.49318 

.86993 

27 

34 

.43156 

.90208 

.44724 

.89441 

.46278 

.88647 

.47818 

.87826 

.49344 

.86978 

26 

35 

.43182 

.90196 

.447.50 

.89428 

.46304 

.88634 

.47844 

.87812 

.49369 

.86964 

25 

36 

.43209 

.90183 

.44776 

.89415 

.46330 

.88620 

.47869 

.87798 

.49394 

.86949 

24 

37 

.43235 

.90171 

.44802 

.89402 

.46355 

.88607 

.47895 

.87784 

.49419 

.86935 

23 

38 

.43261 

.90158 

.44828 

.89389 

.46381 

.88593 

.47920 

.87770 

.49445 

.86921 

22 

39 

.43287 

.90146 

.44854 

.89376 

.46407 

.88580 

.47946 

.87756 

.49470 

.86906 

21 

40 

.43313 

.90133 

.44880 

.89363 

.46433 

.88566 

.47971 

.87743 

.49495 

.86892 

20 

41 

.43340 

.90120 

.44906 

.89350 

.46458 

.88553 

.47997 

.87729 

.49521 

.86878 

19 

42 

.43366 

.90108 

.44932 

.89337 

.46484 

.88539 

.48022 

.87715 

.49546 

.86863 

18 

43 

.43392 

.90095 

.44958 

.89324 

.46510 

.88526 

.48048 

.87701 

.49571 

.86849 

17 

44 

.43418 

.90082 

.44984 

.89311 

.46536 

.88512 

.48073 

.87687 

.49596 

.86834 

16 

45 

.43445 

.90070 

.45010 

.89298 

.46561 

.88499 

.48099 

.87673 

.49622 

.86820 

15 

46 

.43471 

.90057 

.4.5036 

.89285 

.46587 

.88485 

.48124 

.87659 

.49647 

.86805 

14 

47 

.43497 

.90045 

.45062 

.89272 

.46613 

.88472 

.48150 

.87645 

.49672 

.86791 

13 

48 

.43523 

.90032 

.45088 

.89259 

.46639 

.88458 

.48175 

.87631 

.49697 

.86777 

12 

49 

.43549 

.90019 

.45114 

.89245 

.46664 

.88445 

.48201 

.87617 

.49723 

.86762 

11 

50 

.43575 

.90007 

.45140 

.89232 

.46690 

.88431 

.48226 

.87603 

.49748 

.86748 

10 

51 

.43602 

.89994 

.45166 

.89219 

.46716 

.88417 

.48252 

.87589 

.49773 

.86733 

9 

52 

.43628 

.89981 

.45192 

.89206 

.46742 

.88404 

.48277 

.87575 

.49798 

.86719 

8 

63 

.43654 

.89968 

.45218 

.89193 

.46767 

.88390 

.48303 

.87561 

.49824 

.86704 

7 

54 

.43680 

.89956 

.45243 

.89180 

.46793 

.88377 

.48328 

.87546 

.49849 

.86690 

6 

55 

.43706 

.89943 

.45269 

.89167 

.46819 

.88363 

.48354 

.87532 

.49874 

.86675 

5 

56 

.43733 

.89930 

.45295 

.89153 

.46844 

.88349 

.48379 

.87518 

.49899 

.86661 

4 

57 

.43759 

.89918 

.45321 

.89140 

.46870 

.88336 

.48405 

.87504 

.49924 

.86646 

3 

58 

.43785 

.89905 

.45347 

.89127 

.46896 

.88322 

.48430 

.87490 

.49950 

.86632 

2 

59 

.43811 

.89892 

.45373 

.89114 

.46921 

.88308 

.48456 

.87476 

.49975 

.86617 

1 

60 

.43837 

.89879 

.45399 

.89101 

.46947 

.88295 

.48481 

.87462 

.50000 

.86603 

0 

M. 

Cos. 

Sine. 

(’OS. 

Sine. 

Cos. 

Sine. 

Cos. 

Sine. 

Cos. 

Sine. 

M. 


64° 

63° 

62° 

61° 

o 

§ 


































































24 TABLE II. NATURAL SINES AND COSINES, 



30° 

31” 

32° 

33° 

34° 


M. 

Sine. 

Cos. 

Sine. 

Cos. 

Sine. 

Cos. 

Sine.l 

Cos. 

Sine. 

Cos. 

M. 

0 

.50000 

.86603 

.51.504 

.85717 

.52992 

.84805 

.54464 

.83867 

.55919 

.82904 

60 

1 

.50025 

.86588 

.51529 

.85702 

.53017 

.84789 

.54488 

.83851 

.55943 

.82887 

59 

2 

.50050 

.86573 

.51554 

.85687 

.53041 

.84774 

.54513 

.83835 

.55968 

.82871 

58 

3 

.50076 

.86559 

.51579 

.85672 

.5:1066 

.84759 

.54537 

.83819 

.55992 

.82855 

57 

4 

.50101 

.86544 

.51604 

.85657 

.5:4091 

.84743 

.54561 

.83804 

.56016'.82839 

56 

5 

.50126 

.86530 

.51628 

.85642 

.53115 

.84728 

.54586' 

.83788 

.56040 .82822 

55 

6 

.50151 

.86515 

.51653 

.85627 

.53140 

. 84711 ' 

.54610 

.83772 

.56064 .82806 

54 

7 

.50176 

.86501 

.51678 

.85612 

.53164 

.84697 

. 546.’ !5 

.83756 

.56088 

.827110 

53 

8 

.50201 

.86486 

.51703 

.85597 

.53189 

.84681 

.54659 

.83740 

.56112 .82773 

52 

9 

.50227 

.86471 

.51728 

.85582 

.53214 

.84666 

.54683 

.83724 

.56136 

.82757 

51 

10 

.50252 

.86457 

.51753 

.85567 

.53238 

.84650 

.54708 

.83708 

.56160 

.82741 

50 

11 

.50277 

.86442 

.51778 

.85551 

.53263 

.84635 

.54732 

.83692 

.56184 

.82724 

49 

12 

.50302 

.86427 

.51803 

.85536 

.53288 

.84619 

.54756 

.83676 

.56208 

.82708 

48 

13 

.50327 

.86413 

.51828 

.85521 

.53312 

.84604 

.54781 

.83660 

.56232 

1.82692 

47 

14 

.50352 

.86398 

.51852 

.8551 if, 

.53337 

.84588 

.54805 

.83645 

.56256 

.82675 

46 

15 

.50377 

.86384 

.51877 

.85491 

.53361 

•84oi3 

.54829 

.83629 

.56280 

.82659 

45 

16 

.50403 

.86369 

.51902 

.85476 

.53386 

.84557 

.54854 

.83613 

.56305 

j.82643 

44 

17 

.50428 

.86.354 

.51927 

.85461 

.53411 

.84542 

.54878 

.83597 

.56329 

.82626 

43 

18 

.50453 

.86340 

.51952 

.85446 

.53435 

.84526 

.54902 

.83581 

.5G353 

1.82610 

42 

19 

.50478 

.86325 

.51977 

.85431 

.53460 

.84511 

.54927 

.83565 

.56377 

.82593 

41 

20 

.50503 

.86310 

.52002 

.85416 

.53484 

.84495 

.64951 

.83549 

.50401 

1.82577 

40 

21 

.50528 

.86295 

.52026 

.85401 

.53509 

.84480 

.54975 

.83533 

.56425 

.82561 

39 

22 

.50553 

.86281 

.52051 

.85385 

.53534 

.84464 

.54999 

.83517 

.56449 

.82544 

38 

23 

.50578 

.86266; 

.52076 

.85370 

.53558 

.84448 

.55024 

.83501 

.56473 

.82528 

37 

24 

.50603 

.86251 

.52101 

.85355 

.53583 

.84433 

.55048 

.83485 

.56497 

.82511 

36 

25 

.50628 

.86237 

.52126 

.85340 

.53607 

.84417 

.55072 

.83469 

.56521 

.82495 

35 

20 

.50654 

.86222 

.52151 

.85325 

.53632 

.84402 

.55097 

.83153 

.56545 

.82748 

34 

27 

.50679 

.86207 

.52175 

.85310 

.53656 

.84386 

.55121 

.83437 

.56569 

.82462 

33 

28 

.50704 

.86192 

.52200 

.85294 

.53681 

.84370 

.55145 

.83421 

.56593 

.82446 

32 

29 

.50729 

.86178 

.52225 

.85279 

.53705 

.84355 

.55169 

.83405 

.56617 

.82429 

31 

30 

.50754 

.86163 

.52250 

.85264 

.53730 

.84339 

.55194 

.83389 

.56641 

,.82413 

30 

31 

.50779 

.86148 

.52275 

.85249 

.53754 

.84324 

.55218 

.83373 .56665 

.82396 

29 

32 

.50804 

.86133 

.52299 

.85234 

.53779 

.84308 

.65242 

.83356 

.56089 

.82380 

28 

33 

.50829 

.86119 

.52324 

.85218 

.53804 

.84292 

.55266 

.83340 

.56713 

.82363 

27 

34 

.50854 

.86104 

.52349 

.85203 

.53828 

.84277 

.55291 

.8X124 

.56736 

.8234 i 

26 

35 

.50879 

.86089 

.52374 

.85188 

.53853 

.84261 

.55315 

.8X108 

.56760 

.82330 

25 

3<5 

.50904 

.86074 

.52399 

.85173 

.53877 

.84345 

.55339 

.83292 

.56784 

.82314 

24 

37 

.50929 

>6059 

.52423 

.85157 

.53902 

.84230 

.55303 

.83276 

.56808 

.82297 

23 

38 

.50954 

.86045 

.52448 

.85142 

.53926 

.84214 

.55388 

.83260 

.56832 

.82281 

22 

39 

.50979 

.86030 

.52473 

.85127 

.53951 

.84198 

.55412 

.83244 

.56856 

.82264 

21 

40 

.51004 

.86015 

.52498 

.85112 

.53975 

.84182 

.55436 

.83228 

.56880 

.82248 

20 

41 

.51029 

.86000 

.62522 

.85096 

.54000 

.84167 

.55460 

.83212 

.56904 

.82231 

19 

42 

.51054 

.85985 

.52547 

.85081 

.54024 

.84151 

.55484 

.83195 

.56928 

.82214 

18 

43 

.51079 

.85970 

.52572 

.85066 

.54049 

.84135 

.55509 

.83179 

.56952 

.82198 

17 

44 

.51104 

.85956 

.52597 

.85051 

.54073 

.84120 

.55533 

.83163 

.56976 

.82181 

16 

45 

.51129 

.85941 

.52621 

.85035 

.54097 

.84104 

.55557, 

.83147 

.57000 

.82165 

15 

46 

.51154 

.85926 

.52646 

.85020 

.54122 

.84088 

.55581 1 

.83131 

.57024 

.82148 

14 

47 

.51179 

.85911 

.52671 

.85005 

.54146 

.84072 

.55605 

.83115 

.57047 

.82132 

13 

4o 

.51204 

.85896 

.52696 

.84989 

.54171 

.84057 

.55630 

.83098 

.57071 

.82115 

12 

4!> 

.51229 

.85881 

.52720 

.84974 

.54195 

.84041 

.55654 

.83082 

.57095 

.82098 

11 

OH 

.51254 

.85866 

.52745 

.84959 

.54220 

.84025 

.55678 

.83066 

.57119 

.82082 

10 

ol 

.51279 

.85851 

.52770 

.8494:! 

.54244 

.84009 

.55702 

.83050 

.57143 

.82065 

9 


.51304 

.85836 

.52794 

.84928 

.54269 

.83994 

.55726 

.83034 

.57167 

.82048 

8 

oo 

.51329 

.85821 

.52819 

.84913 

.54293 

.83978 

.65750 

.83017 

.57191 

.82032 

rr 

♦ )4 

.51354 

.85806 

.52844 

.84897 

.54317 

.83962 

.55775 

.83001 

.57215 

.82015 

6 

DO 

.51379 

.85792 

.52869 

.84882 

.54342 

.83946 

.55799 

.82985 

.57238 

.81999 

5 

oo 

.51404 

.85777 

.52893 

..84866 

.54366 

.83930 

.55823 

.82969 

.57262 

.81982 

4 

Oi 

.51429 

.85762 

.52918 

.84851 

.54391 

.83915 

.55847 

.82953 

.57286 

.81965 

3 

oo 

.51454 

.85747 

.52943 

.84836 

.54415 

.83899 

.55871 

.82936 

.57310 

.81949 

2 

Oil 

.51479 

.86732 

.52967 

.84820 

.54440 

.83883 

.55895 

.82920 

.57334 

.81932 

1 

oil 

.51504 

.85717 

.52992 

.84805 

.54464 

.83867 

.55919 

.82904 

.57X58 

.81915 

0 

M. 

Cos. 

Sine. 

Cos. 

Sine. 

Cos. 

Sine. 

Cos. 

sine. 

Cos. 

Sine. 

M. 


59o 

58° 

57° 

56° 

55° 

1 

-i 




















































































TABLE II. NATURAL SINES AND COSINES. 25 



35° 

36° 

37° 

w 

00 

o 

39° 


M. 

Sine. 

Cos. 

Sine. 

Cos. 

Pine. 

Cos. 

Sine. 

Cos. 

Sine. 

Cos. 

M. 

0 

.57358 

.81915 

.58779 

.80902 

.60182 

79864 

.61566 

.78801 

.62932 

.77715 

60 

1 

.57381 

.81899 

.58802 

.80885 

.60205 

.79846 

.61589 

.78783 

.62955 

.77696 

59 

2 

.57405 

.81882 

.58826 

.80867 

.60228 

.79829 

.61612 

.78765 

.62977 

.77678 

58 

3 

.57429 

.81865 

.58849 

.80&50 

.60251 

.79811 

.61635 

.78747 

.63000 

.77660 

57 

4 

.57453 

.81848 

.58873 

.80833 

.60274 

.79793 

.61658 

.78729 

.63022 

.77641 

56 

5 

.57477 

.81832 

.58896 

.80816 

.60298 

.79776 

.61681 

.78711 

.63045 

.77623 

55 

G 

.57501 

.81815 

.58920 

.80799 

.60321 

.79758 

.61704 

.78694 

.63068 

.77605 

54 

7 

.57524 

.81798 

.58943 

.80782 

.60344 

.79741 

.61726 

.78676 

.63090 

.77586 

53 

8 

.57548 

.81782 

.58967 

.80765 

.60367 

.79723 

.61749 

.78658 

.63113 

.77568 

52 

9 

.57572 

.81765 

.58990 

.80748 

.60390 

.79706 

.61772 

.78640 

.63135 

.77550 

51 

10 

.57596 

.81748 

.59014 

.80730 

.60414 

.79688 

.61795 

.78622 

.63158 

.77531 

50 

11 

.57619 

.81731 

.59037 

.80713 

.60437 

.79671 

.61818 

.78604 

.63180 

.77513 

49 

12 

.57643 

.81714 

.59061 

.80696 

.60460 

.79653 

.61841 

.78586 

.63203 

.77494 

48 

13 

.57667 

.81698 

.59084 

.80679 

.60483 

.79635 

.61864 

.7856b 

.63225 

.77476 

47 

14 

.57691 

.81681 

.59108 

.80662 

.60506 

.79618 

.61887 

.78550 

.63248 

.77458 

46 

15 

.57715 

.81664 

.59131 

.80644 

.60529 

.79600 

.61009 

.78532 

.63271 

.77439 

45 

16 

.57738 

.81647 

.59154 

.80627 

.60553 

.79583 

.61932 

.78514 

.63293 

.77421 

44 

17 

.57762 

.81631 

.59178 

.80610 

.60576' 

.79565 

.61955 

.78496 

.63316 

.77402 

43 

18 

.57786 

.81614 

.59201 

.80593 

.60599 

.79547 

.61978 

.78478 

.63338 

.77384 

42 

19 

.57810 

.81597 

.59225 

.80576 

.60622 

.79530 

.62001 

.78460 

.63361 

.77366 

41 

20 

.57833 

.81580 

.59248 

.80558 

.60645 

.79512 

.62024 

.78442 

.63383 

.77347 

40 

21 

.57857 

.81563 

.59272 

.80541 

.60668 

.79494 

.62046 

.78424 

.63406 

.77329 

39 

22 

.57881 

.81546 

.59295 

.80524 

.60691 

.79477 

.62069 

.78405 

.63428 

.77310 

38 

23 

.57904 

.81530 

.59318 

.80507 

.60714 

.79459 

.62092 

.78387 

.63451 

.77292 

37 

24 

.57928 

.81513 

.59342 

.80489 

.60738 

.79441 

.62115 

.78369 

.63473 

.77273 

36 

25 

.57952 

.81496 

.59365 

.80472 

.60761 

.79424 

.62138 

.78351 

.63496 

.77255 

35 

26 

.57976 

.81479 

.59389 

.80455 

.60784 

.79406 

.62160 

.78333 

.63518 

.77236 

34 

1 21 

.57999 

.81462 

.59412 

.80438 

.60807 

.79388 

.62183 

.78315 

.63540 

.77218 

33 

\ 28 

.58023 

.81445 

.59436 

.80420 

.60830 

.79371 

.62206 

.78297 

.63563 

.77199 

32 

29 

.58047 

.81428 

.59459 

.80403 

.60853 

.79353 

.62229 

.78279 

.63585 

.77181 

31 

30 

.58070 

.81412 

.59482 

.80386 

.60876 

.79335 

.62251 

.78261 

.63608 

.77162 

30 

31 

.58094 

.81395 

.59506 

.80368 

.60899 

.79318 

.62274 

.78243 

.63630 

.77144 

29 

32 

.58118 

.81378 

.59529 

.80351 

.60922 

.79300 

.62297 

.78225 

.63653 

.77125 

28 

33 

.58141 

.81361 

.59552 

.80334 

.60945 

.79282 

.62320 

.78206 

.63675 

.77107 

27 

34 

.58165 

.81344 

.59576 

.80316 

.60968 

.79264 

.62342 

.78188 

.63698 

.77088 

26 

35 

.58189 

.81327 

.59599 

.80299 

.60991 

.79247 

.62365 

.78170 

.63720 

.77070 

25 

36 

.58212 

.81310 

.59622 

.80282 

.61015 

.79229 

.62388 

.78152 

.63742 

.77051 

24 

37 

.58236 

.81293 

.59646 

.80264 

.61038 

.79211 

.62411 

.78134 

.63765 

.77033 

23 

38 

.58260 

.81276 

.59669 

.80247 

.61061 

.79193 

.62433 

.78116 

.63787 

.77014 

22 

39 

.58283 

.81259 

.59693 

.80230 

.61084 

.79176 

.62456 

.78098 

.63810 

.76996 

21 

40 

.58307 

.81242 

.59716 

.80212 

.61107 

.79158 

•62479 

.78079 

.63832 

.76977 

20 

41 

.58330 

.81225 

.59739 

.80195 

.61130 

.79140 

.62502 

.78061 

.63854 

.76959 

19 

42 

.58354 

.81208 

.59763 

.80178 

.61153 

.79122 

.62524 

.78043 

.63877 

.76940 

18 

43 

.58378 

.81191 

.59786 

.80160 

.61176 

.79105 

.62547 

.78025 

.63899 

.76921 

17 

44 

.58401 

.81174 

.59809 

.80143 

.61199 

.79087 

.62570 

.78007 

.63922 

.76903 

16 

45 

.58425 

.81157 

.59832 

.80125 

!.61222 

.79069 

.62592 

.77988 

1.63944 

.76884 

15 

46 

.58449 

.81140 

.59856 

.80108 

.61245 

.79051 

.62615 

.77970 

1.63966 

.76866 

14 

47 

.58472 

.81123 

.59879 

.80091 

.61268 

.79033 

.62638 

.77952 

.63989 

.76847 

13 

48 

.58496 

.81106 

.59902 

.80073 

.61291 

.79015 

.62660 

.77934 

.64011 

.76828 

12 

49 

■58519 

.81089 

.59926 

.80056 

.61314 

.78998 

1.62683 

.77916 

.64033 

.76810 

11 

50 

.58543 

.81072 

.59949 

.80038 

.61337 

.78980 

.62706 

.77897 

.64056 

.76791 

10 

51 

.58567 

.81055 

.59972 

.80021 

.61360 

.78962 

.62728 

.77879 

.64078 

.76772 

9 

52 

.58590 

.81038 

.59995 

.80003 

.61383 

.78944 

.62751 

.77861 

.64100 

.76754 

8 

63 

.58614 

.81021 

.60019 

.79986 

.61406 

.78926 

.62774 

.77843 

.64123 

.76735 

7 

54 

.58637 

.81004 

.60042 

.79968 

.61429 

.78908 

.62796 

.77824 

.64145 

.76717 

6 

55 

.58661 

.80987 

.60065 

.79951 

.61451 

.78891 

.62819 

.77806 

.64167 

.76698 

5 

56 

.58684 

.80970 

.60089 

.79934 

.61474 

.78873 

.62842 

.77788 

.64190 

.76679 

4 

57 

.58708 

.80953 

.60112 

.79916 

.61497 

.78855 

.62864 

.77769 

.64212 

.76661 

3 

58 

.58731 

.80936 

.60135 

.79899 

.61520 

.78837 

.62887 

.77751 

.64234 

.76642 

2 

59 

.58755 

.80919 

.60158 

.79881 

.61543 

.78819 

.62909 

.77733 

.64256 

.76623 

1 

60 

.58779 

.80902 

.60182 

.79864 

.61566 

.78801 

.62932 

.77715 

.64279 

.76604 

0 

M. 

Cos. 

Sine. 

Cos. 

Sine. 

Cos. 

Sine. 

Cos. 

Sine. 

Cos. 

Sine. 

M. 


54° 

53° 

52° 

51° 

5<r 





















































































26 TABLE II. NATURAL SINES AND COSINES. 



40° 

41 

4 

*1° 

mf 

43° 

44° 


M. 

Sine. 

Cos. 

Sine. 

Cos. 

Sine. 

Cos. 

Sine. 

Cos. 

Sine. 

Cos. 

M. 

0 

64279 

.76604 

.65606 

.75471 

.66913 

.74314 

.68200 

.73135 

.69466 

.71934 

60 

1 

.64301 

.76586 

.65628 

.75452 

.66935 

.74295 

.68221 

.73116 

.69487 

.71914 

59 

2 

.64323 

.76567 

.65650 

.75433 

.66956 

.74276 

.68242 

.73096 

.69508 

.71894 

58 

3 

.64346 

.76548 

.65672 

.77.114 

.66978 

.74256 

.68264 

.73076 

.69529 

.71873 

57 

4 

.64368 

.76530 

.65694 

.75395 

.66999 

.74237 

.68285 

.73056 

.69549 

.71853 

56 

5 

.64390 

.76511 

.65716 

.75375 

.67021 

.74217 

.68306 

.73036 

.69570 

.71833 

55 

6 

.64412 

.76492 

.65738 

.75356 

.67043 

.74198 

68327 

.73016 

.69591 

.71813 

54 

7 

.64435 

.76473 

.65759 

.75337 

.67064 

.74178 

68349 

.72996 

.69612 

.71792 

53 

8 

.64457 

.76455 

.65781 

.75318 

.67086 

.74159 

.68370 

.72976 

.69633 

.71772 

52 

9 

.64479 

.76436 

.65803 

.75299 

.67107 

.74139 

.68391 

.72957 

.69654 

.71752 

51 

10 

.64501 

.76417 

.65825 

.75280 

.67129 

.74120 

.68412 

.72937 

.69675 

.71732 

50 

11 

.64524 

.76398 

.65847 

.75261 

.67151 

.74100 

.684:34 

.72917 

.69696 

.71711 

49 

12 

.64546 

.76380 

.65869 

.75241 

.67172 

.74080 

.68455 

.72897 

.69717 

.71691 

48 

13 

.64568 

.76361 

.65891 

75222 

.67194 

.74061 

.68476 

.72877 

.69737 

.71671 

47 

14 

.64590 

.76342 

.65913 

.75203 

.67215 

.74041 

.68497 

.72857 

.69758 

.71650 

46 

15 

.64612 

.76323 

.65935 

.75184 

.67237 

.74022 

.68518 

.72S37 

.69.79 

.71630 

45 

16 

.64635 

.76304 

.65956 

.75165 

.67258 

.74002 

.68539 

.72817 

.69800 

.71610 

44 

17 

.64657 

.76286 

.65978 

.75146 

.67280 

.73933 

.68561 

.72797 

.69821 

.71590 

43 

18 

.64679 

.76267 

.66000 

.75126 

.67301 

.73963 

.68582 

.72777 

.69842 

.71569 

42 

19 

.64701 

.76248 

.66022 

.75107 

.61323 

.73944 

.68603 

.72757 

.69862 

.71549 

41 

20 

.64723 

.76229 

.66044 

.75088 

.67344 

.73924 

.68624 

.72737 

.69883 

.71529 

40 

21 

.64746 

.76210 

.66066 

.75069 

.67366 

.73904 

.68645 

.72717 

.69904 

.71508 

39 

22 

.64768 

.76192 

.60088 

.75050 

.67387 

.7.1885 

.68666 

.72697 

.69925 

.71488 

38 

23 

.64790 

.76173 

.66109 

.75030 

.67409 

.73865 

.68688 

.72677 

.69940 

.71468 

37 

24 

.64812 

.761.54 

.66131 

.75011 

.67430 

.73846 

. 68709 

.72657 

.69966 

.71447 

36 

25 

.64834 

.76135 

.66153 

.74992 

.67452 

.73826 

.68730 

.72637 

.69987 

.71427 

35 

26 

.64856 

.76116 

.66175 

.74973 

.67473 

.73800 

.68751 

.72617 

.70008 

.71407 

34 

27 

.64878 

.76097 

.66197 

.74953 

.67495 

.73787 

.68772 

.72597 

.70029 

.71386 

33 

28 

.64901 

.76078 

.66218 

.74934 

.67516 

.73767 

.68793 

.72577 

.70049 

.71366 

32 

29 

.61923 

.76059 

.66240 

.74915 

.67538 

.73747 

.68814 

.72557 

.70070 

.71345 

31 

30 

.64945 

.76041 

.66262 

.74896 

.07559 

.73728 

.68835 

.72537 

.70091 

.71325 

30 

31 

.64967 

.76022 

.66284 

.74876 

.67580 

.73708 

.68857 

.72517 

.70112 

.71305 

29 

32 

.64989 

.76003 

.66306 

.74857 

.67602 

.73083 

.68878 

.72497 

.70132 

.71284 

28 

33 

.65011 

.75984 

.66327 

.74838 

.67623 

.73669 

.68899 

.72477 

.70153 

.71264 

27 

34 

.65033 

.75965 

.66349 

.74818 

.67645 

.73049 

.68920 

.72457 

.70174 

.71243 

26 

35 

.65055 

.75946 

.66371 

.74799 

.67666 

.7:3629 

.68941 

.72437 

.70195 

.71223 

25 

36 

.65077 

.75927 

.66393 

.74780 

.67688 

.73610 

.68962 

.72417 

.70215 

.71203 

24 

37 

.65100 

.75908 

.66414 

.74760 

.67709 

.73590 

.68983 

.72397 

.70236 

.71182 

23 

38 

.65122 

.75889 

.66436 

^ 74741 

.67730 

.73570 

.69004 

.72377 

.70257 

.71162 

22 

39 

.65144 

.75870 

.66458 

.74722 

.67752 

.73531 

.69025 

.72357 

.70277 

.71141 

21 

40 

.65166 

.75851 

.66480 

.74703 

.67773 

.73531 

.69040 

.72337 

.70298 

.71121 

20 

41 

.65188 

.75832 

.66501 

.74683 

.67795 

.73511 

.69067 

.72317 

.70319 

.71100 

19 

42 

.65210 

.75813 

.66523 

.74664 

.678i6 

.73491 

. 0! M )S8 

.72297 

.70339 

.71080 

18 

43 

.65232 

.75794 

.60545 

.74644 

.67837 

.73472 

.69109 

.72277 

.70360 

.71059 

17 

44 

.65254 

.75775 

.66566 

.74625 

.67859 

.73452 

.69130 

.72257 

.70381 

.71039 

16 

45 

.65276 

.75756 

.66588 

.74606 

.67880 

.73432 

.69151 

.72236 

.70401 

.71019 

15 

46 

.65298 

.75738 

.66610 

.74586 

.67901 

.73413 

.69172 

.72216 

.70422 

.70998 

14 

47 

.65320 

.75719 

.66632 

.74567 

.67923 

.73393 

.69193 

.72196 

.70443 

.70978 

13 

48 

.65342 

.75700 

.66653 

.74548 

.67944 

.73373 

.69214 

.72176 

.70463 

.70957 

12 

49 

.65364 

.75680 

.66675 

.74528 

.67965 

.73353 

.69235 

.72156 

.70484 

.70937 

11 

50 

.65386 

.75661 

.66697 

.74509 

.67987 

.73333 

.69256 

.72136 

.70505 

.70916 

10 

51 

.65408 

.75642 

.66718 

.74489 

.68008 

.73314 

.69277 

.72116 

.70525 

.70896 

O 

52 

.65430 

.75623 

.66710 

.74470 

.68029 

.73294 

.69298 

.72095 

.70546 

.70875 

8 

53 

.65452 

.75604 

.66762 

.74451 

.68051 

.73274 

.69319 

.72075 

.70567 

.70855 

7 

54 

.65474 

.75585 

.66783 

.74431 

.68072 

.73254 

.69340 

.72055 

.70587 

.70834 

6 

55 

.65496 

.75566 

.66805 

.74412 

.68093 

.73234 

.69361 

.72035 

.70608 

.70813 

5 

56 

.65518 

.75547 

.66827 

.74392 

.68115 

.73215 

.69382 

.72015 

.70628 

.70793 

4 

57 

.65540 

.75528 

.66848 

.74373 

.68136 

.73195 

.69403 

.71995 

.70649 

.70772 

3 

58 

.65562 

.75509 

.66870 

.74353 

.68157 

.73175 

.69424 

.71974 

.70670 

.70752 

2 

59 

.65584 

.75490 

.66891 

.74334 

.6817;. 

.73155 

.69445 

.71954 

.70690 

.70731 

1 

60 

.65606 

.75471 

.66913 

.74314 

.68200 

.73135 

.69466 

.71934 

.70711 

.70711 

0 

_ 1 

M. 

Cos. 

Sine. 

COS. 

l Sine. 

Cos. 

Sine. 

Cos. 

Sine. 

Cos. 

Sine. 

M. 


49 g 

48° 

47° 

46° 

45° 

1 











































































% 

TABLE III. 


NATURAL TANGENTS 

AND 

COTANGENTS. 


127 } 



28 TABLE III. NATURAL TANGENTS, ETC. 



0* 

1° 

O® 

3° 


M 

Tang. 

Cotang 

Tang. 

Cotang 

Tang. 

Cotang 

Tang. 

Cotang 

M* 

0 

.00001 

Infinite 

.01746 

57.290C 

. 03495 

28.6368 

.05241 

19.0811 

60 

1 

.0002! 

3437.75 

.01775 

56.3506 

.03521 

28.3994 

.0,527! 

18.975£ 

69 

2 

.00058 

1718.87 

.01804 

55.4415 

.03550 

28.1664 

.0529! 

18.8711 

68 

3 

.00087 

1145.92 

.01833 

54.5612 

.0357! 

27.9372 

.05326 

18.7678 

57 

4 

.00116 

859.436 

.01862 

53.7086 

.0360! 

27.7117 

.05357 

18.665C 

56 

5 

.00145 

687.549 

.01891 

52.8821 

.0363S 

27.4898 

.05387 

18.564E 

65 

6 

.00175 

572.957 

.01920 

52.086)7 

1 .03667 

27.2715 

.05411 

18.464E 

54 

7 

.00204 

491.106 

.01949 

51.3032 

1 . 03690 

27.0566 

.05445 

18.3655 

53 

8 

.00233 

429.718 

.01978 

50.5485 

.03725 

26.8450 

.0547^ 

18.2677 

62 

9 

.00262 

381.971 

.02007 

49.8157 

.03754 

26.6367 

.05503 

18.1708 

61 

10 

.00291 

343.774 

.02036 

49.1039 

1 .03783 

26.4315 

.05533 

18.0750 

60 

11 

.00320 

312.521 

.02066 

48.4121 

.03812 

26.2296 

.05562 

17.9802 

49 

12 

. 0349 

286.478 

.02095 

47.7395 

.03842 

26.0307 

.05591 

17.8863 

48 

13 

.00378 

264.441 

.02124 

47.0853 

.03871 

25.8348 

.05620 

17.7934 

47 

14 

.00407 

245.552 

.02153 

40.4489 

.03900 

25.6418 

.05649 

17.7015 

46 

15 

.00436 

229.182 

.02182 

45.8294 

.03929 

25.4517 

.05678 

17.6106 

45 

16 

.00465 

214.858 

.02211 

45.2261 

.03958 

25.2644 

.05708 

17.5205 

44 

17 

.00495 

202.219 

.02240 

44.6386 

.03987 

25.0798 

.05737 

17.4314 

43 

18 

.00524 

190.984 

.02269 

44.6)661 

.04016 

24.8978 

.05766 

17.3432 

42 

19 

.00553 

180 932 

.02298 

43.5081 

.04046 

24.7185 

.05795 

17.2558 

41 

20 

.00582 

171.885 

.02328 

42.9641 

.04075 

24.5 418 

.05824 

17.1693 

40 

21 

.00611 

163.700 

.02357 

42.4335 

.04104 

24.3675 

.05854 

17.0837 

39 

22 

.00640 

156.259 

.02386 

41.9158 

.04133 

24.1957 

.05883 

16.9990 

38 

23 

.00669 

149.465 

.02415 

41.4106 

.041621 24.0263 

.05912 

16.9150 

37 

24 

.00698 

143.237 

.02444 

40.9174 

.04191 

, 23.8593 

.05941 

16.8319 

36 

25 

.00727 

137.507 

.02473 

40.4358 

.04220 

| 23.6945 

.05970 

16.7496 

35 

26 

.00756 

132.219 

.02502 

39.9655 

.04250 

23.5321 

.05999 

16.6681 

34 

27 

.00785 

127.321 

.02531 

39.5059 

.04279 

23.3718 

.06029 

16.5874 

33 

28 

.00815 

122.774 

.02560 

39.0568 

.04308 

23.2137 

.06058 

16.5075 

32 

29 

.00844 

118.540 

.02589 

38.6177 

.04337 

23.0577 

.06087 

16.4283 

31 

30 

.00873 

114.589! 

.02619 

38 1885 

.04366 

22.9038 

• 06116 

16.3499 

30 

31 

.00902 

110.892 1 

.02648 

37.7686 

.04395 

22.7519 

.06145 

16.2722 

29 

32 

.00931 

107.426 

.02677 

37.3579 

.04424 

22.6020 

.06175 

16.1952 

28 

33 

.00960 

104.171 

.02706 

36.9560 

.04454 

22.4541 

.06204 

16.1190 

27 

34 

.00989 

101.107 

.02735 

36.5627 

.04483 

22.3081 

.06233 

16.0435 

26 

35 

.01018 

98.2179 

.02764 

36.1776 

.04512 

22.1640 

.06262 

15.9687 

25 

36 

.01047 

95.4895 

.02793 

35.8006 

.04541 

22.0217 

.06291 

15.8945 

24 

37 

.01076 

92.9085 

.02822 

35.4313 

.04570 

21.8813 

.06321 

15.8211 

23 

38 

.01105 

90.4633 

.02851 

35.0695 

.04599 

21.7426 

.06350 

15.7483 

22 

39 

.01135 

88.1436 

.02881 

34.7151 

.04628 

21.6056 

.06379 

15.6762 

21 

40 

.01164 

85.9398 

.02910 

34.3678 

.04658 

21.4704 

.06408 

15.6048 

20 

41 

.01193 

83.8435 

.02939 

34.0273 

.04687 

21.3369 

.06437 

15.5340 

19 

42 

.01222 

81.8470 

.02968 

33.0935 

.04716 

21.2049 

.06467 

15.4638 

18 

43 

.01251 

79.94X4 

.02997 

33.3662 

.04745 

21.0747 

.06490 

15.3943 

17 

44 

.01280 

78.1263 

.03026 

33.0452 

.04774 

20.9460 

.06525 

15.3254 

16 

45 

.01309 

76.3900 

.03055 

32.7303 

.04803 

20.8188 

.06554 

15.2571 

15 

46 

.01338 

74.7292 

.03084 

32.4213 

.04833 

20.6932 

.06584 

15.1893 

14 

47 

.01367 

73.1390 

.03114 

32.1181 

.04862 

20.5691 

.06613 

15.1222 

13 

48 

.01396 

71.6151 

.03143 

31.8205 

.04891 

20.4465 

.06642 

15.0557 

12 

49 

. 1425 

70.1533 

.03172 

31.5284 

.04920 

20.3253 

.06671 

14.9898 

11 

50 

.61455 

68.7501 

.03201 

31.2416 

.0494!) 

20.2056 

.06700 

14.9244 

10 

51 

.01484 

67.4019 

.03230 

30.9599 

.04978 

20.0872 

.06730 

14.8596 

9 

52 

.01513 

66.1055 

.03259 

30.6833 

.05007 

19.9702 

.06759 

14.7954 

8 

53 

.01542 

64.8580 

.03288 

30.4116 

.05037 

19.8.546 

.06788 

14.7317 

7 

54 

.01571 

63.6567 

.03317 

30.1446 

.05066 

19.7403 

.06817 

14.6685 

6 

55 

.01600 

62.4992 

.03346 

29.8823 

.05095 

19.6273 

.06847 

14.6059 

5 

56 

.01629 

61.3829 

.03376 

29.6245 

.05124 

19.5156 

.06876 

14.5438 

4 

57 

.01658 

60.3058 

.03405 

29.3711 

.05153 

19.4051 

.06905 

14.4823 

3 

58 

.01687 

59.2659 

.03434 

29.1220 

.05182 

19.2959 

.06934 

14.4212 

2 

59 

.01716 

58.2612 

.03463 

28.8771 

.05212 

19.1879 

.06963 

14.3607 

1 

60 

.01746 

57.2900 

.03492 

28.63631 

.05241 

19.0811 

.06993 

14.3007 

0 

M. 

Cotang. 

Tang. 

Cotang. | 

Tang. 

Cotang.l 

Tailg. 

Cotang. 

Tang. 

M. 


89° 

88° 

87° 

86° 
























































































TABLE III. NATURAL TANGENTS, ETC. 29 



4° 

5° 

<i° 

7 

o 


M. 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

M. 

0 

.06993 

14.3007 

.08749 

11.4301 

.10510 

9.51436 

.12278 

8.14435 

60 

1 

.07022 

14.2411 

.08778 

11.3919 

.10540 

9.48781 

.12308 

8.12481 

59 

2 

.07051 

14.1821 

.08807 

11.3540 

.10569 

9.46141 

.12338 

8.10536 

58 

3 

.07080 

14.1235 

.08837 

11.3163 

.10599 

9.43515 

.12367 

8.08600 

57 

4 

.07110 

14.0655 

.08866 

11.2789 

.10628 

9.40904 

.12397 

8.06674 

56 

5 

.07139 

14.0079 

.08895 

11.2417 

.10657 

9.38307 

.12426 

8.04756 

55 

G 

.07168 

13.9507 

.08925 

11.2048 

.10687 

9.35724 

.12456 

8.02848 

54 

i 

.07197 

13.8940 

.08954 

11.1681 

.10716 

9.33155 

.12485 

8.00948 

53 

8 

.07227 

13.8378 

.08983 

11.1316 

.10746 

9.30599 

.12515 

7.99058 

52 

9 

.07256 

13.7821 

.09013 

11.0954 

.10775 

9.28058 

.12544 

7.97176 

51 

10 

.07285 

13.7267 

.09042 

11.0594 

.10805 

9.25530 

.12574 

7.95302 

50 

11 

.07314 

13.6719 

.09071 

11.0237 

.10834 

9.23016 

.12603 

7.93438 

49 

12 

.07344 

13.6174 

.09101 

10 9882 

.10863 

9.20516 

.12633 

7.91582 

48 

13 

.07373 

13.5634 

.09130 

10.9529 

.10893 

9.18028 

.12662 

7.89734 

47 

14 

.07402 

13.5098 

.09159 

10.9178 

.10922 

9.15554 

.12692 

7.87895 

46 

15 

.07431 

13.4566 

.09189 

10.8829 

.10952 

9.13093 

.12722 

7.86064 

45 

16 

.07461 

13.4039 

.09218 

10.8483 

.10981 

9.10646 

.12751 

7.84242 

44 

17 

.07490 

13.3515 

.09247 

10.8139 

.11011 

9.08211 

.12781 

7.82428 

43 

18 

.07519 

13.2996 

.09277 

10.7797 

.11040 

9.05789 

.12810 

7.80622 

42 

19 

.07548 

13.2480 

.09306 

10.7457 

.11070 

9.03379 

.12840 

7.78825 

41 

20 

.07578 

13.1969 

.09335 

10.7119 

.11099 

9.00983 

.12869 

7.77035 

40 

21 

.07607 

13.1461 

.09365 

10.6783 

.11128 

8.98598 

.12899 

7.75254 

39 

22 

.07636 

13.0958 

.09394 

10.6450 

.11158 

8.96227 

.12929 

7.73480 

38 

23 

.07665 

13.0458 

.09423 

10.6118 

.11187 

8.93867 

.12958 

7.71715 

37 

24 

.07695 

12.9962 

.09453 

10.5789 

.11217 

8.91520 

.12988 

7.69957 

36 

25 

.07724 

12.9469 

.09482 

10.5462 

.11246 

8.89185 

.13017 

7.68208 

35 

26 

.07753 

12.8981 

.09511 

10.5136 

.11276 

8.86862 

.13047 

7.66466 

34 

27 

.07782 

12.8496 

.09541 

10.4813 

.11305 

8 84551 

.13076 

7.64732 

33 

28 

.07812 

12.8014 

.09570 

10.4491 

.11335 

8.82252 

.13106 

7.63005 

32 

29 

.07841 

12.7536 

.09600 

10.4172 

.11364 

8.79964 

.13136 

7.61287 

31 

30 

.07870 

12.7062 

.09629 

10.3854 

.11394 

8.77689 

.13165 

7.59575 

30 

31 

.07899 

12.6591 

.09658 

10.3538 

.11423 

8.75425 

.13195 

7.57872 

29 

32 

.07929 

12.6124 

.09688 

10.3224 

.11452 

8.73172 

.13224 

7.56176 

28 

33 

.07958 

12.5660 

.09717 

10.2913 

.11482 

8.70931 

.13254 

7.54487 

27 

34 

.07987 

12.5199 

.09746 

10.2602 

.11511 

8.68701 

13284 

7.52806 

26 

35 

.08017 

12.4742 

.09776 

10.2294 

.11541 

8.66482 

.13313 

7.51132 

25 

36 

.08046 

12.4288 

.09805 

10.1988 

.11570 

8.64275 

.13343 

7.49465 

24 

37 

.08075 

12.3838 

.09834 

10.1683 

.11600 

8.62078 

.13372 

7.47806 

23 

38 

.08104 

12.3390 

.09864 

10.1381 

.11629 

8.59893 

.13402 

7.46154 

22 

39 

.08134 

12.2946 

.09893 

10.1080 

.11659 

8.57718 

.13432 

7.44509 

21 

40 

.08163 

12.2505 

.09923 

10.0780 

.11688 

8.55555 

.13461 

7.42871 

20 

41 

.08192 

12.2067 

.09952 

10.0483 

.11718 

8.53402 

.13401 

7.41240 

19 

42 

.08221 

12.1632 

.09981 

10.0187 

.11747 

8.51259 

.13521 

7.39616 

18 

43 

.08251 

12.1201 

.10011 

9.98931 

.11777 

8.49128 

.13550 

7.37999 

17 

44 

.08280 

12.0772 

.10040 

9.90007 

.11806 

8.47007 

.13580 

7.36389 

16 

45 

.08309 

12.0346 

.10069 

9.93101 

.11836 

8.44896 

.13609 

7.34786 

15 

46 

.08339 

11.9923 

.10099 

9.90211 

.11865 

8.42795 

.13639 

7.33190 

14 

47 

.08368 

11.9504 

.10128 

9.87338 

.11895 

8.40705 

.13669 

7.31600 

13 

48 

.08397 

11.9087 

.10158 

9.84482 

.11924 

8.38625 

.13698 

7.30018 

12 

49 

.08427 

11.8673 

.10187 

9.81641 

.11954 

8.36555 

.13728 

7.28442 

11 

50 

.08456 

11.8262 

.10216 

9 78817 

.11983 

8.34496 

.13758 

7 26873 

10 

51 

.08485 

11.7853 

.10246 

9.76009 

.12013 

8.32446 

.13787 

7.25310 

9 

52 

.08514 

11.7448 

.10275 

9.73217 

.12042 

8.30406 

.13817 

7.23754 

8 

53 

.08544 

11.7045 

.10305 

9 70441 

.12072 

8.28376 

.13846 

7.22204 

7 

54 

.08573 

11.6645 

.10334 

9.67680 

.12101 

8.26355 

.13876 

7.20661 

6 

55 

.08602 

11.6248 

.10363 

9.64935 

.12131 

8.24345 

.13906 

7.19125 

5 

56 

.08632 

11.5853 

.10393 

9.62205 

.12160 

8.22344 

.13935 

7.17594 

4 

57 

.08661 

11.5461 

.10422 

9.59490 

.12190 

8.20352 

.13965 

7.16071 

3 

58 

.08690 

11.5072 

.10452 

9.56791 

.12219 

8.18370 

.13995 

7.14553 

2 

59 

.08720 

11.4685 

.10481 

9.54106 

.12249 

8.16398 

.14024 

7.13042 

1 

60 

.08749 

11.4301 

.10510 

9.51436 

.12278 

8.14435 

.14054 

7.11537 

0 

M. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

M. 


85° 

84° 

83° 

82° 

























































30 TABLE III. NATURAL TANGENTS, ETC. 



8° 

9 

S 

0 

11° 


M- 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

M. 

0 

.14054 

7.11537 

.15838 

6.31375 

.17633 

6.67128 

.19438 

5.14455 

60 

1 

.14084 

7.10038 

.15868 

6.30189 

.17663 

5.66165 

.19468 

5.13658 

69 

2 

.14113 

7.08.546 

.15898 

6.29007 

.17693 

6.65205 

.19498 

5.12862 

58 

3 

.14143 

7.07059 

.15928 

6.27829 

.17723 

5.64248 

.19529 

5.12069 

57 

4 

.14173 

7.05579 

.15958 

6.26655 

.17753 

5.63295 

.19559 

5.11279 

r >6 

5 

.14202 

7.04105 

.15988 

6.25486 

.17783 

5.62344 

.19589 

5.10490 

55 

6 

.14232 

7.02637 

.16017 

6.24321 

.17813 

5.61397 

.19619 

5.09704 

54 

7 

.14262 

7.01174 

.16047 

6.23160 

.17843 

5.60452 

.19649 

5.08921 

53 

8 

.14291 

G.99718 

.16077 

6.22003 

.17873 

5.59511 

.19680 

5.08139 

52 

P 

.14321 

6.98268 

.16107 

6.20851 

.17903 

5.58573 

.19710 

5.07360 

51 

10 

.14351 

G.96823 

.16137 

6.19703 

.17933 

5.57638 

.19740 

5.06584 

50 

11 

.14381 

G.95385 

.16167 

6.18559 

.17963 

5.56706 

.19770 

5.05809 

49 

12 

.14410 

G.93952 

.16196 

6.17419 

.17993 

5.55777 

.19801 

5.05037 

48 

13 

.14440 

0.92525 

.16226 

6.16283 

.18023 

5.54851 

.19831 

5.04267 

47 

14 

.14470 

G.91104 

.16256 

6.15151 

.18053 

5.53927 

.19861 

5.03499 

46 

15 

.14499 

6.89688 

.16286 

6.14023 

.18083 

5.53007 

.19891 

5.02734 

45 

16 

.14529 

G.88278 

.16316 

6.12899 

.18113 

5.52090 

.19921 

5.01971 

44 

17 

.14559 

6.86874 

.16346 

6.11779 

.18143 

6.51176 

.19952 

5.01210 

43 

18 

.14588 

6.85475 

.16376 

6.10664 

.18173 

6.50264 

.19982 

5.00451 

42 

19 

.14618 

6.84082 

.16405 

6.09552 

.18203 

5.49.356 

.20012 

4.99695 

41 

20 

.14648 

6.82694 

.16435 

6.08444 

.18233 

5.48451 

.20042 

4.98940 

40 

21 

.14678 

6.81312 

.16465 

6.07340 

.18263 

5.47548 

.20073 

4.98188 

39 

22 

.14707 

6.79936 

.16495 

6.06240 

.18293 

5.46648 

.20103 

4.97438 

38 

23 

.14737 

6.78564 

.16525 

6.05143 

.18323 

6.45751 

.20133 

4.96690 

37 

24 

.14767 

6.77199 

.16555 

6.04051 

.18&53 

5.44857 

.20164 

4.95945 

30 

25 

.14796 

6.75838 

.16585 

6.02962 

.18384 

5.43966 

.20194 

4.95201 

35 

26 

.14826 

6.74483 

.16615 

6.01878 

.18414 

5.43077 

.20224 

4.94460 

34 

27 

.14856 

6.73133 

.16645 

6.00797 

.18444 

5.42192 

.20254 

4.93721 

33 

28 

.14886 

6.71789 

.16674 

5.99720 

.18474 

5.41309 

.20285 

4.92984 

32 

29 

.14915 

6.70450 

.16704 

5.98646 

.18.504 

5.40429 

.20315 

4.92249 

31 

30 

.14945 

6.69116 

.16734 

5.97576 

.18534 

5.39552 

.20345 

4.91516 

30 

31 

.14975 

6.67787 

.16764 

5.96510 

.18564 

5.38677 

.20376 

4.90785 

29 

32 

.15005 

6.66463 

.16794 

5.95448 

.18594 

5.37805 

.20406 

4.90056 

28 

33 

.15034 

6.65144 

.16824 

5.94390 

.18624 

5.36936 

.20436 

4.89330 

27 

34 

.15064 

6.63831 

.16854 

5.93335 

.18654 

5.36070 

.20466 

4.88605 

26 

35 

.15094 

6.62523 

.16884 

5.92283 

.18684 

5.35206 

.20497 

4.87882 

25 

36 

.15124 

6.61219 

.16914 

5.91236 

.18714 

5.34345 

.20527 

4.87162 

24 

37 

.15153 

6.59921 

.16944 

5.90191 

.18745 

5.33487 

.20557 

4.86444 

23 

38 

.15183 

6.58627 

.16974 

5.89151 

.18775 

5.32631 

.20588 

4.85727 

22 

39 

.15213 

6.57339 

.17004 

5.88114 

.18805 

5.31778 

.20618 

4.85013 

21 

40 

.15243 

6.56055 

.17033 

5.87080 

.18835 

5.30928 

.20648 

4.84300 

20 

41 

.15272 

6.54777 

.17063 

5.86051 

.18865 

5.30080 

.20679 

4.83590 

19 

42 

.15302 

6.53503 

.17093 

5.85024 

.18895 

5.29235 

.20709 

4.82882 

18 

43 

.15332 

6.52234 

.17123 

5.84001 

.18925 

5.28393 

.20739 

4.82175 

17 

44 

.15362 

6.50970 

.17153 

5.82982 

.18955 

5.27553 

.20770 

4 81471 

16 

45 

.15391 

6.49710 

.17183 

5 81966 

.18986 

5.26715 

.20800 

4.80769 

15 

46 

.1.5421 

6.48456 

.17213 

5.80953 

.19016 

5.25880 

.20830 

4.80068 

14 

47 

.15451 

6.47206 

.17243 

5.79944 

.19046 

5.25048 

.20861 

4.79370 

13 

48 

.15481 

6.45961 

.17273 

5.78938 

.19076 

5.24218 

.20891 

4.78673 

12 

49 

.15511 

6.44720 

.17303 

5.77936 

.19106 

5.23391 

.20921 

4.77978 

11 

50 

.15540 

6.43484 

.17333 

5.76937 

.19136 

5.22566 

.20952 

4.77286 

10 

51 

.15570 

6.42253 

.17363 

5.75941 

.19166 

5.21744 

.20982 

4.76595 

9 

52 

.15600 

6.41026 

.17393 

5.74949 

.19197 

5.20925 

.21013 

4.75906 

8 

53 

.15830 

6.39804 

.17423 

5.73960 

.19227 

5.20107 

.21043 

4.75219 

7 • 

54 

.15360 

6.38587 

.17453 

5.72974 

.19257 

5.19293 

.21073 

4.74534 

6 

55 

.15689 

6.37374 

.17483 

5.71992 

.19287 

5.18480 

.21104 

4.73851 

5 

56 

.15719 

6.36165 

.17513 

5.71013 

.19317 

5.17671 

.21134 

4.73170 

4 

57 

.15749 

6.34961 

.17.543 

5.70037 

.19347 

5.16863 

.21164 

4.72490 

3 

68 

.15779 

6.33761 

.17573 

5.69064 

.19378 

5.16058 

.21195 

4.71813 

2 

69 

.15809 

6.32566 

.17603 

5.68094 

.19408 

5.15256 

.21225 

4.71137 

1 

60 

.15838 

6.31375 

.17633 

5.67128 

.19438 

5.14455 

.21256 

4.70463 

0 

M. 

Co tang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

M. 


81* 

oc 

o 

o 

79° 

78° 



























































TABLE III. NATURAL TANGENTS, ETC 


31 



12° 

i:r 

14° 

15° 


M. 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

M. 

0 

.21256 

4.70463 

.23087 

4.33148 

.24933 

4.01078 

.26795 

3.73205 

60 

1 

.21286 

4.69791 

.23117 

4.32573 

.24964 

4.00582 

.26826 

3.72771 

59 

2 

.21316 

4.69121 

.23148 

4.32001 

.24995 

4.00086 

.26857 

3.72338 

58 

3 

.21347 

4.68452 

.23179 

4.31430 

.25026 

3.99592 

.26888 

3.71907 

57 

4 

.21377 

4.67786 

.•23209 

4.30860 

.25056 

3.99099 

.26920 

3.71476 

56 

5 

.21408 

4.67121 

.23240 

4.30291 

.25087 

3.98607 

.26951 

3.71046 

55 

6 

.21438 

4.66458 

.23271 

4.29724 

.25118 

3.98117 

.26982 

3.70616 

54 

7 

.21469 

4.65797 

.23301 

4.29159 

.25149 

3.97627 

.27013 

3.70188 

53 

8 

.21499 

4.65138 

.23332 

4.28595 

.25180 

3.97139 

.27044 

3.69761 

52 

9 

.21529 

4.64480 

.23363 

4.28032 

.25211 

3.96651 

.27076 

3.69335 

51 

JO 

.21560 

4.63825 

.23393 

4.27471 

.25242 

3.96165 

.27107 

3.68909 

50 

11 

.21590 

4.63171 

.23424 

4.26911 

.25273 

3.95680 

.27138 

3.68485 

49 

12 

.21621 

4.62518 

.23455 

4.26352 

.25304 

3.95196 

.27169 

3.68061 

48 

13 

.21651 

4.61868 

.23485 

4.25795 

.25335 

3.94713 

.27201 

3.67638 

47 

14 

.21682 

4.61219 

.23516 

4.25239 

.25366 

3.94232 

.27232 

3.67217 

46 

15 

.21712 

4.60572 

.23547 

4.24685 

.25397 

3.93751 

.27263 

3.66796 

45 

16 

.21743 

4.59927 

.23578 

4.24132 

.25428 

3.93271 

.27294 

3.66376 

44 

17 

.21773 

4.59283 

.23608 

4.23580 

.25459 

3 92793 

.27326 

3.65957 

43 

18 

.21804 

4.58641 

.23639 

4.23030 

.25490 

3.92316 

.27357 

3.65538 

42 

19 

.21834 

4.58001 

.23670 

4.22481 

.25521 

3.91839 

.27388 

3.65121 

41 

20 

.21864 

4.57363 

.23700 

4.21933 

.25552 

3.91364 

.27419 

3.64705 

40 

21 

.21895 

4.56726 

.23731 

4.21387 

.25583 

3.90890 

.27451 

3.64289 

39 

22 

.21925 

4.56091 

.23762 

4.20842 

.25614 

3.90417 

.27482 

3.63874 

38 

23 

.21956 

4.55458 

.23793 

4.20298 

.25645 

3.89945 

.27513 

3.63461 

37 

24 

.21986 

4.54826 

.23823 

4.19756 

.25676 

3.89474 

.27545 

3.63048 

36 

25 

.22017 

4.54196 

.23854 

4.19215 

.25707 

3.89004 

.27576 

3.62636 

35 

26 

.22047 

4.53568 

. 2.3885 

4.18675 

.25738 

3.88536 

.27607 

3.62224 

34 

27 

.22078 

4.52941 

.23916 

4.18137 

.25769 

3.88068 

.27638 

3.61814 

33 

28 

.22108 

4.52316 

.23946 

4.17600 

.25800 

3.87601 

.27670 

3.61405 

32 

29 

.22139 

4.51693 

.23977 

4.17064 

.25831 

3.87136 

.27701 

3.60996 

31 

30 

.22169 

4.51Q71 

.24008 

4.16530 

.25862 

3.86671 

.27732 

3.60588 

30 

31 

.22200 

4.50451 

.24039 

4.15997 

.25893 

3.86208 

.27764 

3.60181 

29 

32 

.22231 

4.49832 

.24069 

4.15465 

.25924 

3.85745 

.27795 

3.59775 

28 

33 

.22261 

4.49215 

.24100 

4.14934 

.25955 

3.85284 

.27826 

3.59370 

27 

34 

.22292 

4.48600 

.24131 

4.14405 

.25986 

3.84824 

.27858 

3.58966 

26 

35 

.22322 

4.47986 

.24162 

4.13877 

.26017 

3.84364 

.27889 

3.58562 

25 

36 

.22353 

4.47374 

.24193 

4.13350 

.26048 

3.83906 

.27921 

3.58160 

24 

37 

.22383 

4.46764 

.24223 

4.12825 

.26079 

3.83449 

.27952 

3.57758 

23 

38 

.22414 

4.46155 

.24254 

4.12301 

.26110 

3.82992 

.27983 

3.57357 

22 

39 

.22444 

4.45548 

.24285 

4.11778 

.26141 

3.82537 

.28015 

3.56957 

21 

40 

.22475 

4.44942 

.24316 

4.11256 

.26172 

3.82083 

.28046 

3.56557 

20 

41 

.22505 

4.44338 

.24347 

4.10736 

.26203 

3.81630 

.28077 

3.56159 

19 

42 

.22536 

4.43735 

.24377 

4.10216 

.26235 

3.81177 

.28109 

3.55761 

18 

43 

.22567 

4.43134 

.24408 

4.09699 

.26266 

3.80726 

.28140 

3.55364 

17 

44 

.22597 

4.42534 

.24439 

4.09182 

.26297 

3.80276 

.28172 

3.54968 

16 

45 

.22628 

4.41936 

.24470 

4.08666 

.26328 

3.79827 

.28203 

3.54573 

15 

46 

.22658 

4.41340 

.24501 

4.08152 

.26359 

3.79378 

.28234 

3.54179 

14 

47 

.22689 

4.40745 

.24532 

4.07639 

.26390 

3.78931 

.28266 

3.53785 

13 

48 

.22719 

4.40152 

.24562 

4.07127 

.26421 

3.78485 

.28297 

3.53393 

12 

49 

.22750 

4.39.560 

.24593 

4.06616 

.26452 

3.78040 

.28329 

3.53001 

11 

50 

.22781 

4.38969 

.24624 

4.06107 

.26483 

3.77595 

.28360 

3.52609 

10 

51 

.22811 

4.38381 

.24655 

4.05599 

.26515 

3.77152 

.28391 

3.52219 

9 

52 

.22842 

4.37793 

.24686 

4.05092 

.26546 

3.76709 

.28423 

3.51829 

8 

53 

.22872 

4.37207 

.24717 

4.04586 

.26577 

3.76268 

.28454 

3.51441 

7 

54 

.22903 

4.36623 

.24747 

4.04081 

.26608 

3.75828 

.28486 

3.51053 

6 

55 

.22934 

4.36040 

.24778 

4.03578 

.26639 

3.75388 

.28517 

3.50666 

5 

56 

.22964 

4.35459 

.24809 

4.03076 

.26670 

3.749.50 

.28549 

3.50279 

4 

57 

.22995 

4.34879 

.24840 

4.02574 

.26701 

3.74512 

.28580 

3.49894 

3 

58 

.23026 

4.34300 

.24871 

4.02074 

.26733 

3.74075 

.28612 

3.49509 

2 

59 

.23056 

4.33723 

.24902 

4.01576 

.26764 

3.73640 

.28643 

3.49125 

1 

60 

.23087 

4.33148 

.24933 

4.01078 

.26795 

3.73205 

.28675 

3.48741 

0 

M. 

Cotang. 

Tailg. 

Cotang. 

Tang, 

Cotang. 

Tang. 

Cotang. 

Tailg. 

M. 


77° 

76° 

7S° 

74° 













































32 TABLE III. NATURAL TANGENTS, ETC. 



16° 

17° 

18° 

19° 


M. 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Co tang. 

Tang. 

Cotang. 

M. 

0 

.28675 

3.48741 

.30573 

3.27085 

.32492 

3.077681 

.34433 

2.90421 

60 

1 

.28706 

3.48359 

.30605 

3.26745 

.32524, 

3.07464: 

.34465 

2.90147 

59 

2 

.28738 

3.47977 

.30637 

3.26406 

.32556 

3.07160! 

.34498 

2.89873! 58 

3 

.28769 

3.47596 

.30669 

3.26067 

.32588 

3.06857! 

.34530 

2.89600 

57 

4 

.28800 

3.47216 

.30700 

3.25729 

.32621 

3.06554, 

.34563 

2.89327 

56 

5 

.28832 

3.46837 

.30732 

3.25392 

.32653 

3.062521 

.34596 

2.89055 

55 

G 

.28864 

3.46458 

.30764 

3.25055 

.32685 

3.059501 

.34628 

2.88783 

54 

7 

.28895 

3.46080 

.30796 

3.24719 

.32717 

3.056491 

.34661 

2.88511 

53 

8 

.28927 

3.45703 

.30828 

3.24383 

.32749 

3.05349' 

.34693 

2.88240 

52 

9 

.28958 

3.45327 

.30860 

3.24049 

.32782 

3.050491 

.34726 

2.87970 

51 

10 

.28990 

3.44951 

.30891 

3.23714 

.32814 

3.047491 

.34758 

2.87700 

50 

11 

.29021 

3.44576 

.30923 

3.23381 

.32846 

3.04450] 

.34791 

2.87430 

49 

12 

.29053 

3.44202 

.30955 

3.23048 

.32878 

3.04152; 

.34824 

2.87161 

48 

13 

.29084 

3.43829 

.30987 

3.22715 

.32911 

3.03854 

.34856 

2.86892 

47 

14 

.29116 

3.43456 

.31019 

3.22384 

.32943 

3.03556! 

.34889 

2.86624 

46 

15 

.29147 

3 .43084 

.31051 

3.22053 

.32975 

3.03260) 

.34922 

2.86356 

45 

16 

.29179 

3.42713 

.31083 

3.21722 

.33007 

3.02963 

.34954 

2.86089 

44 

17 

.29210 

3.42343 

.31115 

3.21392 

.33040 

3.02667 

.34987 

2.85822 

43 

18 

.29242 

3.41973 

.31147 

3.21063 

.33072 

3.02372 

.35020 

2.85555 

42 

19 

.29274 

3.41604 

.31178 

3.20734 

.33104 

3.02077 

.35052 

2.85289 

41 

20 

.29305 

3.41236 

.31210 

3.20406 

.33136] 

3.01783 

.35085 

2.85023 

40 

21 

.29337 

3.40869 

.31242 

3.20079 

.33169, 

3.01489 

.35118 

2.84758 

39 

22 

.29368 

3.40502 

.31274 

3.19752 

.33201 

3.01196 

.35150 

2.84494 

38 

23 

.29400 

3.40136 

.31306 

3.19426 

.33233 

3.00903 

.35183 

2.84229 

37 

24 

.29432 

3.39771 

.31338 

3.19100 

.33266 

3.00611 

.35216 

2.83965 

36 

25 

.29463 

3.39406 

.31370 

3.18775 

.33298 

3.00319 

.35248 

2.83702 

35 

26 

.29495 

3.39042 

.31402 

3.18451 

.33330 

3.00028 

.35281 

2.83439 

34 

27 

.29526 

3.38679 

.31434 

3.18127 

.33363 

2.99738 

.35314 

2.83176 

33 

28 

.29558 

3.38317 

.31466 

3.17804 

.33395 

2.99447 

.35346 

2.82914 

32 

29 

.29590 

3.37955 

.31498 

3.17481 

.33427 

2.99158 

.35379 

2.82653 

31 

30 

.29621 

3.37594 

.31530 

3.17159 

.33460 

2.98868 

.35412 

2.82391 

30 

31 

.29653 

3.37234 

.31562 

3.16838 

.33492 

2.98580 

.3.5445 

2.82130 

29 

32 

.29685 

3.36875 

.31594 

3.16517 

.33524 

2.98292 

.35477 

2.81870 

28 

33 

.29716 

3.36516 

.31626 

3.16197 

.33557 

2.98004 

.35510 

2.81610 

27 

34 

.29748 

3.36158 

.31658 

3.15877 

.33580 

2.97717 

.35543 

2.81350 

26 

35 

.29780 

3.35800 

.31690 

3.15558 

.33621 

2.97430 

. 35576 

2.81091 

25 

36 

.29811 

3.35443 

.31722 

3.15240 

.33654 

2.97144 

.35608 

2.80833 

24 

37 

.29843 

3.35087 

.31754 

3.14922 

.33686 

2.96858 

.35641 

2.80574 

23 

38 

.29875 

3.34732 

.31786 

3.14605 

.33718 

2.96573 

.35674 

2.80316 

22 

39 

.29906 

3.34377 

.31818 

3.14288 

.33751 

2.96288 

.35707 

2.80059 

21 

40 

.29938 

3.34023 

.31850. 

3.13972 

.33783 

2.96004 

.35740 

2.79802 

20 

41 

.29970 

3.33670 

.31882 

3.13656 

.33816 

2.95721 

.35772 

2.79545 

19 

42 

.30001 

3.33317 

.31914 

3.13341 

.33848 

2.95437 

.35805 

2.79289 

18 

43 

.30033 

3.32965 

.31946 

3.13027 

.33881 

2.95155 

.35838 

2.79033 

17 

44 

.30065 

3.32614 

.31978 

3.12713 

.33913 

2.94872 

.35871 

2.78778 

16 

45 

.30097 

3.32264 

.32010 

3.12400 

.33945 

2.94591 

.35904 

2.78523 

15 

46 

.30128 

3.31914 

.32042 

3.12087 

.33978 

2.94309 

.35937 

2.78269 

14 

47 

.30160 

3.31565 

.32074 

3.11775 

.34010 

2.94028 

.35969 

2.78014 

13 

48 

.30192 

3.31216 

.32106 

3.11464 

.34043 

2.93748 

.36002 

2.77761 

12 

49 

.30224 

3.30868 

.32139 

3.11153 

.34075 

2.93468 

.36035 

2.77507 

11 

50 

.30255 

3.30521 

.32171 

3.10842 

.34108 

2.93189 

.36068 

2.77254 

10 

51 

.30287 

3.30174 

.32203 

3.10532 

.34140 

2.92910 

.36101 

2.77002 

9 

52 

.30319 

3.29829 

.32235 

3.10223 

.34173 

2.92632 

.36134 

2.76750 

8 

53 

.30351 

3.29483 

.32267 

3.09914 

.34205 

2.92354 

.36167 

2.76498 

7 

54 

.30382 

3.29139 

.32299 

3.09606 

.34238 

2.92076 

.36199 

2.76247 

6 

55 

.30414 

3.28795 

.32331 

3.09298 

.34270 

2.91799 

.36232 

2.75996 

5 

56 

.30446 

3.28452 

.32363 

3.08991 

.34303 

2.91523 

.36265 

2.75746 

4 

57 

.30478 

3.28109 

.32396 

3.08685 

.34335 

2.91246 

.36298 

2.75496 

3 

58 

.30509 

3.27767 

.32428 

3.08379 

.34368 

2.90971 

.36331 

2.75246 

2 

59 

.30541 

3.27426 

.32460 

8.08073 

.34400 

2.90696 

.36364 

2.74997 

i 

60 

.30573 

3.27085 

.32492 

3.07768 

.34433 

2.90421 

.36397 

2.74748 

0 

M 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

M. 


73° 

72° 

71° 

70° 



































































TABLE III. NATURAL TANGENTS, ETC 


33 



o 

0* 

21 

L° 


2 



M. 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

M. 

0 

.36397 

2.74748 

.38386 

2.60509 

.40403 

2.47509 

.42447 

2.35585 

60 

1 

.36430 

2.74499 

.38420 

2.60283 

.40436 

2.47302 

.42482 

2 35395 

59 

2 

.36463 

2.74251 

.38453 

2.60057 

.40470 

2.47095 

.42516 

2.35205 

58 

3 

.36496 

2.74004 

.38487 

2.59831 

.40504 

2.46888 

.42551 

2.35015 

57 

4 

.36529 

2.73756 

.38520 

2.59606 

.40538 

2.46682 

.42585 

2.34825 

56 

5 

.36562 

2.73509 

.38553 

2.59381 

.40572 

2.46476 

.42619 

2.34636 

55 

6 

.36595 

2.73263 

.38587 

2.59156 

.40606 

2.46270 

.42654 

2.34447 

54 

7 

.36628 

2.73017 

.38620 

2.58932 

.40640 

2.46065 

.42688 

2.34258 

53 

8 

.36661 

2.72771 

.38654 

2.58708 

.40674 

2.45860 

.42722 

2.34069 

52 

9 

.36694 

2.72526 

.38687 

2.58484 

.40707 

2.45655 

.42757 

2.33881 

51 

10 

.36727 

2.72281 

.38721 

2.58261 

.40741 

2.45451 

.42791 

2.33693 

50 

11 

.36760 

2.72036 

.38754 

2.58038 

.40775 

2.45246 

.42826 

2.33505 

49 

12 

.36793 

2.71792 

.38787 

2.57815 

.40809 

2.45043 

.42860 

2.33317 

48 

13 

.36826 

2.71548 

.38821 

2.57593 

.40843 

2.44839 

.42894 

2.33130 

47 

14 

.36859 

2.71305 

.38854 

2.57371 

.40877 

2.44636 

.42929 

2.32943 

46 

15 

.36892 

2.71062 

.38888 

2.57150 

.40911 

2.44433 

.42963 

2.32756 

45 

16 

.36925 

2.70819 

.38921 

2.56928 

.40945 

2.44230 

.42998 

2.32570 

44 

17 

.36958 

2.70577 

.38955 

2.56707 

.40979 

2.44027 

.43032 

2.32383 

43 

18 

.36991 

2.70335 

.38988 

2.56486 

.41013 

2.43825 

.43067 

2.32197 

42 

19 

.37024 

2.70094 

.39022 

2.56266 

.41047 

2.43623 

.43101 

2.32012 

41 

20 

.37057 

2.69853 

.39055 

2.56046 

.41081 

2.43422 

.43136 

2.31826 

40 

21 

.37090 

2.69612 

.39089 

2.55827 

.41115 

2.43220 

.43170 

2.31641 

39 

22 

.37123 

2.69371 

.39122 

2.55608 

.41149 

2.43019 

.43205 

2.31456 

38 

23 

.37157 

2.69131 

.39156 

2.55389 

.41183 

2.42819 

.43239 

2.31271 

37 

24 

.37190 

2.68892 

.39190 

2.55170 

.41217 

2.42618 

.43274 

2.31086 

36 

25 

.37223 

2.68653 

.39223 

2.54952 

.41251 

2.42418 

.43308 

2.30902 

35 

26 

.37256 

2.68414 

.39257 

2.54734 

.41285 

2.42218 

.43343 

2.30718 

34 

27 

.37289 

2.68175 

.39290 

2.54516 

.41319 

2.42019 

.43378 

2.30534 

33 

28 

.37322 

2.67937 

.39324 

2.54299 

.41353 

2.41819 

.43412 

2.30351 

32 

29 

.37355 

2.67700 

.39357 

2.54082 

.41387 

2.41620 

.43447 

2.30167 

31 

30 

.37388 

2.67462 

.39391 

2.53865 

.41421 

2.41421 

.43481 

2.29984 

30 

31 

.37422 

2.67225 

.39425 

2.53618 

.41455 

2.41223 

.43516 

2.29801 

29 

32 

.37455 

2.66989 

.39458 

2.53432 

.41490 

2.41025 

.43550 

2.29619 

28 

33 

.37488 

2.66752 

.39492 

2.53217 

.41524 

2.40827 

.43585 

2.29437 

27 

34 

.37521 

2.66516 

.39526 

2.53001 

.41558 

2.40629 

.43620 

2.29254 

26 

35 

.37554 

2.66281 

.39559 

2.52786 

.41592 

2.40432 

.43654 

2.29073 

25 

36 

.37588 

2.66046 

.39593 

2.52571 

.41626 

2.40235 

.43689 

2.28891 

24 

37 

.37621 

2.65811 

.39626 

2.52357 

.41660 

2.40038 

43724 

2.28710 

23 

38 

.37654 

2.65576 

.39660 

2.52142 

.41694 

2.39841 

.43758 

2.28528 

22 

39 

.37687 

2.65342 

39694 

2.51929 

.41728 

2.39645 

.43793 

2.28348 

21 

40 

.37720 

2.65109 

.39727 

2.51715 

.41763 

2.39449 

.43828 

2.28167 

20 

41 

.37754 

2.64875 

.39761 

2.51502 

.41797 

2.39253 

.43862 

2.27987 

19 

42 

.37787 

2.64642 

.39795 

2.51289 

.41831 

2.39058 

.43897 

2.27806 

18 

43 

.37820 

2.64410 

.39829 

2.51076 

.41865 

2.38863 

.43932 

2.27626 

17 

44 

.37853 

2.64177 

.39862 

2.50864 

.41899 

2.38688 

.43966 

2.27447 

16 

45 

.37887 

2.63945 

.39896 

2.50652 

.41933 

2.38473 

.44001 

2.27267 

15 

46 

.37920 

2.63714 

.39930 

2.50440 

.41968 

2.38279 

.44036 

2.27088 

14 

47 

.37953 

2.63483 

.39963 

2.50229 

.42002 

2 38084 

.44071 

2.26909 

13 

48 

.37986 

2.63252 

.39997 

2.50018 

.42036 

2.37891 

.44105 

2.26730 

12 

49 

.38020 

2.6:3021 

.40031 

2.49807 

.42070 

2.37697 

.44140 

2.26552 

11 

50 

.38053 

2.62791 

.40065 

2.49597 

.42105 

2.37504 

.44175 

2.26374 

10 

51 

.38086 

2.62561 

.40098 

2.49386 

.42139 

2.37311 

.44210 

2.26196 

9 

52 

.38120 

2.62332 

.40132 

2.49177 

.42173 

2.37118 

.44244 

2.26018 

8 

53 

.38153 

2.62103 

.40166 

2.48967 

.42207 

2.36925 

.44279 

2.25840 

7 

54 

.38186 

2.61874 

.40200 

2.48758 

.42242 

2.36733 

.44314 

2.25663 

6 

55 

.38220 

2.61646 

.40234 

2 48549 

.42276 

2.36541 

,44349 

2.25486 

5 

56 

.38253 

2.61418 

.40267 

2.48340 

.42310 

2.36349 

.44384 

2.25309 

4 

57 

.38286 

2.61190 

.40301 

2.48132 

.42345 

2.36158 

.44418 

2.25132 

3 

58 

.38320 

2.60963 

.40335 

2.47924 

.42379 

2.35967 

.44453 

2.24956 

2 

59 

.38353 

2.60736 

.40369 

2.47716 

.42413 

2.35776 

.44488 

2.24780 

1 

60 

.38386 

2.60509 

.40403 

2.47509 

.42447 

2.35585 

.44523 

2.24604 

0 

M. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

M. 


i 

o 

50 

68° 

67° 

66° 




















































34 TABLE III. NATURAL TANGENTS, ETC. 



24* 

Of** 

26° 

27° 


M 

. Tang. 

Cotang 

Tang. 

Cotang 

Tang. 

Cotang 

Tang. 

Cotang 

. M. 

( 

) . 4452.' 

5 2.24601 

.46631 

2.14451 

.48773 

i 2.0503( 

.50953 

1 1.96261 

60 

1 

.44551“ 

2.24428 

. 4666( 

2.1428!- 

.48801 

> 2.04871 

.5098! 

* 1.96121 

) 59 

c 

.4459£ 

2.24252 

.4670* 

2.14127 

.48843 

2.0472? 

. 51021 

1.9597! 

) 58 


.44627 

2.24077 

.46737 

2.13963 

.48881 

2.04577 

.51063 

1.9583? 

57 

4 

.4466: 

2.23902 

.4677: 

2.13801 

.48917 

2.04421 

.5109£ 

1.9569? 

56 

£ 

.44697 

2.23727 

.46808 

2.13639 

.48953 

2.04271 

.51131 

1.95551 

55 

C 

.44732 

2.23553 

.46843 

2.13477 

.48989 

2.04123 

.51173 

1.95417 

54 

7 

.44767 

2.23378 

.46879 

2.133K 

,49021i 

2.03975 

.5120S 

1.95277 

53 

8 

.44802 

2.23201 

.46914 

2.13154 

.49062 

2.03825 

.51246 

1.95137 

52 

9 

.44837 

2.23030 

.46950 

2.12993 

.49098 

2.03675 

.51283 

1.94997 

51 

10 

.44872 

2.22857 

.46985 

2.12832 

.49134 

2.03526 

.51319 

1.94858 

50 

11 

.44907 

2.22683 

.47021 

2.12671 

.49170 

2.03376 

.51356 

1.94718 

49 

12 

.44942 

2.22510 

.47056 

2.12511 

.49206 

2.03227 

.51393 

1.94579 

48 

13 

.44977 

2.22337 

.47092 

2.12350 

.49242 

2.03078 

.51430 

1.94440 

47 

14 

.4.5012 

2.22164 

.47128 

2.12190 

.49278 

2.02929 

.51467 

1.94301 

46 

15 

.45047 

2.21992 

.47163 

2.12030 

.49315 

2.02780 

.51503 

1.94162 

45 

16 

.45082 

2.21819 

.47199 

2.11871 

.49351 

2.02631 

.51540 

1.94023 

44 

17 

.45117 

2.21647 

.47234 

2.11711 

.49387 

2.02483 

.51577 

1.93885 

43 

18 

.45152 

2.21475 

.47270 

2.11552 

.49423 

2.02335 

.51614 

1.93746 

42 

19 

•.45187 

2.21304 

.47305 

2.11392 

.49459 

2.02187 

.51651 

1.93608 

41 

20 

.45222 

2.21132 

.47341 

2.11233 

.49495 

2.02039 

.51688 

1.93470 

40 

21 

.45257 

2.20961 

.47377 

2.11075 

.49532 

2.01891 

.51724 

1.93332 

39 

22 

.45292 

2.20790 

.47412 

2.10916 

.49568 

2.01743 

.51761 

1.93195 

38 

23 

.45327 

2.20619 

.47448 

2.10758 

.49604 

2.01596 

.51798 

1.93057 

37 

24 

.45362 

2.20449 

.47483 

2.10600 

.49640 

2.01449 

.51835 

1.92920 

36 

25 

.45397 

2.20278 

.47519 

2.10442 

.49677 

2.01302 

.51872 

1.92782 

35 

26 

.45432 

2.20108 

.47555 

2.10284 

.49713 

2.01155 

.51909 

1.92645 

34 

27 

.45467 

2.19938 

.47590 

2.10126 

.49749 

2.01008 

.51946 

1.92508 

33 

28 

.45502 

2.19769 

.47626 

2.09969 

.49786 

2.00862 

.51983 

1.92371 

32 

29 

.45538 

2.19599 

.47662 

2.09811 

.49822 

2.00715 

.52020 

1.92235 

31 

30 

.45573 

2.19430 

.47698 

2.09654 

.49858 

2.00569 

.52057 

1.92098 

30 

31 

.45608 

2.19261 

.47733 

2.09498 

.49894 

2.00423 

.52094 

1.91962 

29 

32 

.45643 

2.19092 

.47769 

2.09341 

.49931 

2.00277 

.52131 

1.91826 

28 

33 

.45678 

2.18923 

.47805 

2.09184 

.49967 

2.00131 

.52168 

1.91690 

27 

34 

.45713 

2.18755 

.47840 

2.09028 

.50004 

1.99986 

.52205 

1.91554 

26 

35 

.45748 

2.18587 

.47876 

2.08872 

.50040 

1.99841 

.52242 

1.91418 

25 

36 

.45784 

2.18419 

.47912 

2.08716 

.50076 

1.99695 

.52279 

1.91282 

24 

37 

.45819 

2.18251 

.47948 

2.08560 

.50113 

1.99550 

.52316 

1.91147 

23 

38 

.45854 

2.18084 

.47984 

2.08405 

.50149 

1.99406 

.52353 

1.91012 

22 

39 

.45889 

2.17916 

.48019 

2.08250 

.50185 

1.99261 

.52390 

1.90876 

21 

40 

.45924 

2.17749 

.48055 

2.08094 

.50222 

1.99116 

.52427 

1.90741 

20 

41 

.45960 

2.17582 

.48091 

2.07939 

.50258 

1.98972 

.52464 

1.90607 

19 

42 

.45995 

2.17416 

.48127 

2.07785 

.50295 

1.98828 

.52501 

1.90472 

18 

43 

.46030 

2.17249 

.48163 

2.07630 

.50331 

1.98684 

.52538 

1.903137 

17 

44 

.46065 

2.17083 

.48198 

2.07476 

.50368 

1.98540 

.52575 

1.90203 

16 

45 

.46101 

2.16917 

.48234 

2.07321 

.50404 

1.98396 

.52613 

1.90069 

15 

46 

.46136 

2.16751 

.48270 

2.07167 

.50441 

1.98253 

.52650 

1.89935 

14 

47 

.46171 

2.16585 

.48306 

2.07014 

.50477 

1.98110 

.52687 

1.89801 

13 

48 

.46206 

2.16420 

.48342 

2.06860 

.50514 

1.97966 

.52724 

1.89667 

12 

49 

.46242 

2.16255 

.48378 

2.06706 

.50550 

1.97823 

.52761 

1.89533 

11 

50 

.46277 

2.16090 

.48414 

2.06553 

.50587 

1.97681 

.52798 

1.89400 

10 

51 

.46312 

2.15925 

.48450 

2.06400 

.50623 

1.97538 

.52836 

1.89266 

9 

52 

.46348 

2.15760 

.48486 

2.06247 

.50660 

1.97395 

.52873 

1.89133 

8 

53 

.46383 

2.15596 

.48521 

2.06094 

.50696 

1.97253 

.52910 

1.89000 

7 

54 

.46418 

2.15432 

.48557 

2.05942 

.50733 

1.97111 

.52947 

1.88867 

6 

55 

.46454 

2.15268 

.48593 

2.05790 

.50769 

1.96969 

.52985 

1.88734 

5 

56 

.46489 

2.15104 

.48629 

2.05637 

.50806 

1.96827 

.53022 

1.88602 

4 

57 

.46525 

2.14940 

.48665 

2.05485 

.50843 

1.96685 

.53059 

1.88469 

3 

58 

.46560 

2.14777 

.48701 

2.05333 

.50879 

1.96544 

.53096 

1.88337 

2 

59 

.46595 

2.14614 

.48737 

2.05182 

.50916 

1.96402 

.53134 

1.88205 

1 

60 

.46631 

2.14451 

.48773 

2.05030 

.50953 

1.96261 

.53171 

1.88073 

0 

M. 

Cotang. 

Tang. 

Cotang 

Tang. 

Cotang. 

Tang. 

Cotang 

Tang. 

M. 


65° 

04° 

63° 

62° 








































































TABLE III. NATURAL TANGENTS, ETC. 35 


r 

28° 

29° 

30° 

31° 


M. 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

M. 

0 

.53171 

1.88073 

.55431 

1.80405 

.57735 

1.73205 

.60086 

1.66428 

60 

1 

.53208 

1.87941 

.55469 

1.80281 

.57774 

1.73089 

.60126 

1.66318 

59 

2 

.53246 

1.87809 

.55507 

1.80158 

.57813 

1.72973 

.60165 

1.66209 

58 

3 

.53283 

1.87677 

.55545 

1.80034 

.57851 

1.72857 

.60205 

1.66099 

57 

4 

.53320 

1.87546 

.55583 

1.79911 

.57890 

1.72741 

.60245 

1.65990 

56 

5 

.53358 

1.87415 

.55621 

1.79788 

.57929 

1.72625 

.60284 

1.65881 

55 

6 

.53395 

1.87283 

.55659 

1.79665 

.57968 

1.72509 

.60324 

1.65772 

54 

7 

.53432 

1.87152 

.55697 

1.79542 

.58007 

1.72393 

.60364 

1.65663 

53 

8 

.53470 

1.87021 

.55736 

1.79419 

.58046 

1.72278 

.60403 

1.65554 

52 

9 

.53507 

1.86891 

.55774 

1.79296 

.58085 

1.72163 

.60443 

1.65445 

51 

10 

.53545 

1.86760 

.55812 

1.79174 

.58124 

1.72047 

.60483 

1.65337 

50 

11 

.53582 

1.86630 

.55850 

1.79051 

.58162 

1.71932 

.60522 

1.65228 

49 

12 

.53620 

1.86499 

.55888 

1.78929 

.58201 

1.71817 

.60562 

1.65120 

48 

13 

.53657 

1.86369 

.55926 

1.78807 

.58240 

1.71702 

.60602 

1.65011 

47 

14 

.53694 

1.86239 

.55964 

1.78685 

.58279 

1.71588 

.60642 

1.64903 

46 

15 

.53732 

1.86109 

.56003 

1 78563 

.58318 

1.71473 

.60681 

1.64795 

45 

16 

.53769 

1.85979 

.56041 

1.78441 

.58357 

1.71358 

.60721 

1.64687 

44 

17 

.53807 

1.85850 

.56079 

1.78319 

.58396 

1.71244 

.60761 

1.64579 

43 

18 

.53844 

1.85720 

.56117 

1.78198 

.58435 

1.71129 

.60801 

1.64471 

42 

19 

.53882 

1.85591 

.56156 

1.78077 

.58474 

1.71015 

.60841 

1.64363 

41 

20 

.53920 

1.85462 

.56194 

1.77955 

.58513 

1.70901 

.60881 

1.64256 

40 

21 

.53957 

1.85333 

.56232 

1.77834 

.58552 

1.70787 

.60921 

1.64148 

39 

22 

.53995 

1.85204 

.56270 

1.77713 

.58591 

1.70673 

.60960 

1.64041 

38 

23 

.54032 

1.85075 

.56309 

1.77592 

.58631 

1.70560 

.61000 

1.63934 

37 

24 

.54070 

1.84946 

.56347 

1.77471 

.58670 

1.70446 

.61040 

1.63826 

36 

25 

.54107 

1.84818 

.56385 

1.77351 

.58709 

1.70332 

.61080 

1.63719 

35 

26 

.54145 

1.84689 

.56424 

1.77230 

.58748 

1.70219 

.61120 

1.63612 

34 

27 

.54183 

1.84561 

.56462 

1.77110 

.58787 

1.70106 

.61160 

1.63505 

33 

28 

.54220 

1.84433 

.56501 

1.76990 

.58826 

1.69992 

.61200 

1.63398 

32 

29 

.54258 

1.84305 

.56539 

1.76869 

.58865 

1.69879 

.61240 

1.63292 

31 

30 

.54296 

1.84177 

.56577 

1.76749 

.58905 

1.69766 

.61280 

1.63185 

30 

31 

.54333 

1.84049 

.56616 

1.76629 

.58944 

1.69653 

.61320 

1.63079 

29 

32 

.54371 

1.83922 

.56654 

1.76510 

.58983 

1.69541 

.61360 

1.62972 

28 

33 

.54409 

1.83794 

.56693 

1.76390 

.59022 

1.69428 

.61400 

1.62866 

27 

34 

.54446 

1.83667 

.56731 

1.76271 

.59061 

1.69316 

.61440 

1.62760 

26 

35 

.54484 

1.83540 

.56769 

1.76151 

.59101 

1.69203 

.61480 

1.62654 

25 

36 

.54522 

1.83413 

.56808 

1.76032 

.59140 

1.69091 

.61520 

1.62548 

24 

37 

.54560 

1.83286 

.56846 

1.75913 

.59179 

1.68979 

.61561 

1.62442 

23 

38 

.54597 

1.83159 

.56885 

1.75794 

.59218 

1.68866 

.61601 

1.62336 

22 

39 

.54635 

1.83033 

.56923 

1.75675 

.59258 

1.68754 

.61641 

1.62230 

21 

40 

.54673 

1.82906 

.56962 

1.75556 

.59297 

1.68643 

.61681 

1.62125 

20 

41 

.54711 

1.82780 

.57000 

1.75437 

.59336 

1.68531 

.61721 

1.62019 

19 

42 

.54748 

1.82654 

.57039 

1.75319 

.59376 

1.68419 

.61761 

1.61914 

18 

43 

.54786 

1.82528 

.57078 

1.75200 

.59415 

1.68308 

.61801 

1.61808 

17 

44 

.54824 

1.82402 

.57116 

1.7.5082 

.59454 

1.68196 

.61842 

1.61703 

16 

45 

.54862 

1.82276 

.57155 

1.74964 

.59494 

1.68085 

.61882 

1.61598 

15 

46 

.54900 

1.82150 

.57193 

1.74846 

.59533 

1.67974 

.61922 

1.61493 

14 

47 

.54938 

1.82025 

.57232 

1.74728 

.59573 

1.67863 

.61962 

1.61388 

13 

48 

.54975 

1.81899 

.57271 

1.74610 

.59612 

1.67752 

.62003 

1.61283 

12 

49 

.55013 

1.81774 

.57309 

1.74492 

.59651 

1.67641 

.62043 

1.61179 

11 

50 

.55051 

1.81649 

.57348 

1.74375 

.59691 

1.67530 

.62083 

1.61074 

10 

51 

.55089 

1.81524 

.57386 

1.74257 

.59730 

1.67419 

.62124 

1.60970 

9 

52 

.55127 

1.81399 

.57425 

1.74140 

.59770 

1.67309 

.62164 

1.60865 

8 

53 

.55165 

1.81274 

.57464 

1.74022 

.59809 

1.67198 

.62204 

1.60761 

7 

54 

.55203 

1.81150 

.57503 

1.73905 

.59849 

1.67088 

.62245 

1.60657 

6 

55 

.55241 

1.81025 

.57541 

1.73788 

.59888 

1.66978 

.62285 

1.60553 

5 

56 

.55279 

1.80901 

.57580 

1.73671 

.59928 

1.66867 

.62325 

1.60449 

4 

57 

.55317 

1.80777 

.57619 

1.73555 

.59967 

1.66757 

.62366 

1.60345 

8 

58 

.55355 

1.80653 

.57657 

1.73438 

.60007 

1.66647 

.62406 

1.60241 

2 

59 

.55393 

1.80529 

.57696 

1.73321 

.60046 

1.66538 

.62446 

1.60137 

1 

60 

.55431 

1.80405 

.57735 

1.73205 

.60086 

1.66428 

.62487 

1.60033 

0 

M. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

M. 


61° 

o 

o 

o 

— 

59° 

58° 






























































86 TABLE III. NATURAL TANGENTS, ETC. 



3 

O o 

/V 

33° 

o 

CO 

i 

o 

to 

00 


M- 

Tang. 

Cotang 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

M- 

0 

.62487 

1.60033 

.64941 

1.53986 

.67451 

1.48256 

.70021 

1.42815 

60 

1 

.62527 

1.59930 

.64982 

1.53888 

.67493 

1.48163 

.70064 

1.42726 

59 

2 

.62568 

1.59826 

.65024 

1.53791 

.67536 

1.48070 

.70107 

1.42638 

58 

3 

.62608 

1.59723 

.65065 

1.53693 

.67578 

1.47977 

.70151 

1.42550 

57 

4 

.62649 

1.59620 

.65100 

1.53595 

.67620 

1.47885 

.70194 

1.42462 

56 

f> 

.62689 

1.59517 

.65148 

1.53497 

.67663 

1.47792 

.70238 

1.42374 

55 

6 

.62730 

1.59414 

.65189 

1.53400 

.67705 

1.47699 

.70281 

1.42286 

54 

7 

.62770 

1.59311 

.65231 

1.53302 

.67748 

1.47607 

.70325 

1.42198 

53 

8 

.62811 

1.59208 

.65272 

1.53205 

.67790 

1.47514 

.70368 

1.42110 

52 

y 

.62852 

1.59105 

.65314 

1.53107 

.67832 

1.47422 

.70412 

1.42022 

51 

10 

.62892 

1.59002 

.65355 

1.53010 

.67875 

1.47330 

.70455 

1.41934 

50 

n 

.62933 

1.58900 

.65397 

1.52913 

.67917 

1.47238 

.70499 

1.41847 

49 

12 

.62973 

1.58797 

.65438 

1.52816 

.67960 

1.47146 

.70542 

1.41759 

48 

13 

.63014 

1.58695 

.65480 

1.52719 

.68002 

1.47053 

.70586 

1.41672 

47 

14 

.63055 

1.58593 

.65521 

1.52622 

.68045 

1.46962 

.70629 

1.41584 

46 

15 

.63095 

1.58490 

.65563 

1.52525 

.68088 

1.46870 

.70673 

1.41497 

45 

16 

.63136 

1.58388 

.65604 

1.52429 

.68130 

1.46778 

.70717 

1.41409 

44 

17 

.63177 

1.58286 

.65646 

1.52332 

.68173 

1.46686 

.70760 

1.41322 

43 

18 

.63217 

1.58184 

.65688 

1.52235 

.68215 

1.46595 

.70804 

1.41235 

42 

19 

.63258 

1.58083 

.65729 

1.52139 

.68258 

1.46503 

.70848 

1.41148 

41 

20 

.63299 

1.57981 

.65771 

1.52043 

.68301 

1.46411 

.70891 

1.41061 

40 

21 

.63340 

1.57879 

.65813 

1.51946 

.68343 

1.46320 

.70935 

1.40974 

39 

22 

.63380 

1.57778 

.65854 

1.51850 

.68386 

1.46229 

.70979 

1.40887 

38 

23 

.63421 

1.57676 

.65896 

1.51754 

.68429 

1.46137 

.71023 

1.408C0 

37 

24 

.63462 

1.57575 

.65938 

1.51658 

.68471 

1.46046 

.71066 

1.40714 

36 

25 

.63503 

1.57474 

.65980 

1.51562 

.68514 

1.45955 

.71110 

1.40627 

35 

26 

.63544 

1.57372 

.66021 

1.51466 

.68557 

1.45864 

.71154 

1.40540 

34 

27 

.63584 

1.57271 

.66063 

1.51370 

.68600 

1.45773 

.71198 

1.40454 

33 

28 

.63625 

1.57170 

.66105 

1.51275 

.68642 

1.45682 

.71242 

1.40367 

32 

29 

.63666 

1.57069 

.66147 

1.51179 

.68685 

1.45592 

.71285 

1.40281 

31 

30 

.63707 

1.56969 

.66189 

1.51084 

.68728 

1.45501 

.71329 

1.40195 

30 

31 

.63748 

1.56868 

.66230 

1.50988 

.68771 

1.45410 

.71373 

1.40109 

29 

32 

.63789 

1.56767 

.66272 

1.50893 

.68814 

1.45320 

.71417 

1.40022 

28 

33 

.63830 

1.56667 

.66314 

1.50797 

.68857 

1.45229 

.71461 

1.39936 

27 

34 

.63871 

1.56566 

.66356 

1.50702 

.68900 

1.45139 

.71505 

1.39850 

26 

35 

.63912 

1.56466 

.66398 

1.50607 

.68942 

1.45049 

.71549 

1.39764 

25 

36 

.63953 

1.56366 

.66440 

1.50512 

.68985 

1.44958 

.71593 

1.39679 

24 

37 

.63994 

1.56265 

.66482 

1.50417 

.69028 

1.44868 

.71637 

1.39593 

23 

38 

.64035 

1.56165 

.66524 

1.50322 

.69071 

1.44778 

.71681 

1.39507 

22 

39 

.64076 

1.56065 

.66566 

1.50228 

.69114 

1.44688 

.71725 

1.39421 

21 

40 

.64117 

1.55966 

.66608 

1.50133 

.69157 

1.44598 

.71769 

1.39336 

20 

41 

.64158 

1.55866 

.66650 

1.50038 

.69200 

1.44508 

.71813 

1.39250 

19 

42 

.64199 

1.55766 

.66692 

1.49944 

.69243 

1.44418 

.71857 

1.39165 

18 

43 

.64240 

1.55666 

.66734 

1.49849 

.69286 

1.44329 

.71901 

1.39079 

17 

44 

.64281 

1.55567 

.66776 

1.49755 

.69329 

1.44239 

.71946 

1.38994 

16 

45 

.64322 

1.55467 

.66818 

1.49661 

.69372 

1.44149 

.71990 

1.38909 

15 

46 

.64363 

1.55368 

.66860 

1.49566 

.69416 

1.44060 

.72034 

1.38824 

14 

47 

.64404 

1.55269 

.66902 

1.49472 

.69459 

1.43970 

.72078 

1.38738 

13 

48 

.64446 

1.55170 

.66944 

1.49378 

.69502 

1.43881 

.72122 

1.38653 

12 

49 

.64487 

1.55071 

.66986 

1.49284 

.69545 

1.43792 

.72167 

1.38568 

11 

50 

.64528 

1.54972 

.67028 

1.49190 

.69588 

1.43703 

.72211 

1.38484 

10 

51 

.64569 

1.54873 

.67071 

1.49097 

.69631 

1.43614 

.72255 

1.38399 

9 

52 

.64610 

1.54774 

.67113 

1.49003 

.69675 

1.43525 

.72299 

1.38314 

8 

53 

.64652 

1.54675 

.67155 

1.48909 

.69718 

1.43436 

.72344 

1.38229 

7 

54 

.64693 

1.54576 

.67197 

1.48816 

.69761 

1.43347 

.72388 

1.38145 

6 

55 

.64734 

1.54478 

.67239 

1.48722 

.69804 

1.43258 

.72432 

1.38060 

5 

56 

.64775 

1.54379 

.67282 

1.48629 

.69847 

1.43169 

.72477 

1.37976 

d 

57 

.64817 

1.54281 

.67324 

1.48536 

.69891 

1.43080 

.72521 

1.37891 

3 

58 

.64858 

1.54183 

.67366 

1.48442 

.69934 

1.42992 

.72565 

1.37807 

2 

59 

.64899 

1.54085 

.67409 

1.48349 

.69977 

1.42903 

.72610 

1.37722 

1 

60 

.64941 

1.53986 

.67451 

1.48256 

.70021 

1.42815 

.72654 

1.37638 

0 

M. 

Cotang. 

Tang. 

Cotang. 1 

Tang. 

Cotang. 

Tang. 

Cotang. | 

Tang. 

M. 


5 

7• 

56° 1 

55° 

54° 








































































TABLE III. NATURAL TANGENTS, ETC. 37 




37° 

38° 

39° 


M. 

Tang. 

Co tang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

M- 

0 

.72654 

1.37638 

.75355 

1.32704 

.78129 

1.27994 

.80978 

1.23490 

60 

1 

.72699 

1.37554 

.75401 

1.32624 

.78175 

1.27917 

.81027 

1.23416 

59 

2 

.72743 

1.37470 

.75447 

1.32544 

.78222 

1.27841 

.81075 

1.23343 

58 

3 

.72788 

1.37386 

.75492 

1.32464 

.78269 

1.27764 

.81123 

1.23270 

57 

4 

.72832 

1.37302 

.75538 

1.32384 

.78316 

1.27688 

.81171 

1.23196 

56 

5 

.72877 

1.37218 

.75584 

1.32304 

.78363 

1.27611 

.81220 

1.23123 

55 

6 

.72921 

1.37134 

.75629 

1.32224 

.78410 

1.27535 

.81268 

1.23050 

54 

7 

.72966 

1.37050 

.75675 

1.32144 

.78457 

1.27458 

.81316 

1.22977 

53 

8 

.73010 

1.36967 

.75721 

1.32064 

.78504 

1.27382 

.81364 

1.22904 

52 

9 

.73055 

1.36883 

.75767 

1.31984 

.78551 

1.27306 

.81413 

1.22831 

51 

10 

.73100 

1.36800 

.75812 

1.31904 

.78598 

1.27230 

.81461 

1.22758 

50 

11 

.73144 

1.36716 

.75858 

1.31825 

.78645 

1.27153 

.81510 

1.22685 

49 

12 

.73189 

1.36633 

.75904 

1.31745 

.78692 

1.27077 

.81558 

1.22612 

48 

13 

.73234 

1.36549 

.75950 

1.31666 

.78739 

1.27001 

.81606 

1.22539 

47 

14 

.73278 

1.36466 

.75996 

1.31586 

.78786 

1.26925 

.81655 

1.22467 

46 

15 

.73323 

1.36383 

.76042 

1.31507 

.78834 

1.26849 

.81703 

1.22394 

45 

16 

.73368 

1.36300 

.76088 

1.31427 

.78881 

1.26774 

.81752 

1.22321 

44 

17 

.73413 

1.36217 

.76134 

1.31348 

.78928 

1.26698 

.81800 

1.22249 

43 

18 

.73457 

1.36134 

.76180 

1.31269 

.78975 

1.26622 

.81849 

1.22176 

42 

19 

.73502 

1.36051 

.76226 

1.31190 

.79022 

1.26546 

.81898 

1.22104 

41 

20 

.73547 

1.35968 

.76272 

1.31110 

.79070 

1.26471 

.81946 

1.22031 

40 

21 

.73592 

1.35885 

.76318 

1.31031 

.79117 

1.26395 

.81995 

1.21959 

39 

22 

.73637 

1.35802 

.76364 

1.30952 

.79164 

1.26319 

.82044 

1.21886 

38 

23 

.73681 

1.35719 

.76410 

1.30873 

.79212 

1.26244 

.82092 

1.21814 

37 

24 

.73726 

1.35637 

.76456 

1.30795 

.79259 

1.26169 

.82141 

1.21742 

36 

25 

.73771 

1.35554 

.76502 

1.30716 

.79306 

1.26093 

.82190 

1.21670 

35 

26 

.73816 

1.35472 

.76548 

1.30637 

.79354 

1.26018 

.82238 

1.21598 

34 

27 

.73861 

1.35389 

.76594 

1.30558 

.79401 

1.25943 

.82287 

1.21526 

33 

28 

.73906 

1.35307 

.76640 

1.30480 

.79449 

1.25867 

.82336 

1.21454 

32 

29 

.73951 

1.35224 

.76686 

1.30401 

.79496 

1.25792 

.82385 

1.21382 

31 

30 

.73996 

1.35142 

.76733 

1.30323 

.79544 

1.25717 

.82434 

1.21310 

30 

31 

.74041 

1.35060 

.76779 

1.30244 

.79591 

1.25642 

.82483 

1.21238 

29 

32 

.74086 

1.34978 

.76825 

1.30166 

.79639 

1.25567 

.82531 

1.21166 

28 

33 

.74131 

1.34896 

.76871 

1.30087 

.79686 

1.25492 

.82580 

1.21094 

27 

34 

.74176 

1.34814 

.76918 

1.30009 

.79734 

1.25417 

.82629 

1.21023 

26 

35 

.74221 

1.34732 

.76964 

1.29931 

.79781 

1.25343 

.82678 

1.20951 

25 

36 

.74267 

1.34650 

.77010 

1.29853 

.79829 

1.25268 

.82727 

1.20879 

24 

37 

.74312 

1.34568 

.77057 

1.29775 

.79877 

1.25193 

.82776 

1.20808 

23 

38 

.74357 

1.34487 

.77103 

1.29696 

.79924 

1.25118 

.82825 

1.20736 

22 

39 

.74402 

1.34405 

.77149 

1.29618 

.79972 

1.25044 

.82874 

1.20665 

21 

40 

.74447 

1.34323 

.77196 

1.29541 

.80020 

1.24969 

.82923 

1.20593 

20 

41 

.74492 

1.34242 

.77242 

1.29463 

.80067 

1.24895 

.82972 

1.20522 

19 

42 

.74538 

1.34160 

.77289 

1.29385 

.80115 

1.24820 

.83022 

1.20451 

18 

43 

.74583 

1.34079 

.77335 

1.29307 

.80163 

1.24746 

.83071 

1.20379 

17 

44 

.74628 

1.33998 

.77382 

1.29229 

.80211 

1.24672 

.83120 

1.20308 

16 

45 

.74674 

1.33916 

.77428 

1.29152 

.80258 

1.24597 

.83169 

1.20237 

15 

46 

.74719 

1.33835 

.77475 

1.29074 

.80306 

1.24523 

.83218 

1.20166 

14 

47 

.74764 

1.33754 

.77521 

1.28997 

.80354 

1.24449 

.83268 

1.20095 

13 

48 

.74810 

1.33673 

.77568 

1.28919 

.80-102 

1.24375 

.83317 

1.20024 

12 

49 

.74855 

1.33592 

.77615 

1.28842 

.80150 

1.24301 

.83366 

1.19953 

11 

50 

.74900 

1.33511 

.77661 

1.28764 

.80498 

1.24227 

.83415 

1.19882 

10 

51 

.74946 

1.33430 

.77708 

1.28687 

.80546 

1.24153 

.83465 

1.19811 

9 

52 

.74991 

1.33349 

.77754 

1.28610 

.80594 

1.24079 

.83514 

1.19740 

8 

53 

.75037 

1.33268 

.77801 

1.28533 

.80642 

1.24005 

.83564 

1.19669 

7 

54 

.75082 

1.33187 

.77848 

1.28456 

.80690 

1.23931 

.836J3 

1.19599 

6 

55 

.75128 

1.33107 

.77895 

1.28379 

.80738 

1.23858 

.83662 

1.19528 

5 

56 

.75173 

1.33026 

.77941 

1.28302 

.80786 

1.23784 

.83712 

1.19457 

4 

57 

.75219 

1.32946 

.77988 

1.28225 

.80834 

1.23710 

.83761 

1.19387 

3 

58 

.75264 

1.32865 

.78035 

1.28148 

.80882 

1.23637 

.83811 

1.19316 

2 

59 

.75310 

1.32785 

.78082 

1.28071 

.80930 

1.23563 

.83860 

1.19246 

1 

60 

.75355 

1.32704 

.78129 

1.27994 

.80978 

1.23490 

.83910 

1.19175 

0 

*1 

Cotang. 

Tang. 

Cotang. 

1 Tang, 

Cotang. 

Tang. 

Cotang. 

Tang. 

M. 



52° 

51° 

50° 



















































38 TABLE III. NATURAL TANGENTS, ETC. 



© 

o 

41° 

>4 

a 

o 

__ 1 

43° 


M. 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

M. 

0 

.83010 

1.19175 

.86929 

1.15037 

.90040 

1.11061 

.93252 

1.07237 

60 

1 

.83060 

1.19105 

.86980 

1.14969 

.90093 

1.10996 

.93306 

1.07174 

59 

2 

.84000 

1.19035 

.87031 

1.14902 

.90146 

1.10931 

.93360 

1.07112 

58 

3 

.84050 

1.18961 

.87082 

1.14834 

.90199 

1.10867 

.93415 

1.07049 

57 

4 

.84108 

1.18894 

.87133 

1.14767 

. 90251 

1.10802 

.93469 

1.06987 

56 

5 

.84158 

1.18824 

.87184 

1.14699 

.90304 

1.10737 

.93524 

1.06925 

55 

C 

.84208 

1.18754 

.87236 

1.14632 

.90357 

1.10672 

.93578 

1.06862 

54 

7 

.84258 

1.18684 

.87287 

1.14565 

.90410 

1.10607 

.93633 

1.06800 

53 

8 

.84307 

1.18614 

.87338 

1.14498 

.90463 

1.10543 

.93688 

1.06738 

52 

y 

.84357 

1.18514 

.87389 

1.14430 

.90516 

1.10478 

.93742 

1.06676 

51 

10 

.84407 

1.18474 

.87441 

1.14363 

.90569 

1.10414 

.93797 

1.06613 

50 

li 

.84457 

1.18404 

.87402 

1.14296 

.90621 

1.10349 

.93852 

1.06551 

49 

12 

.84507 

1.18334 

.87543 

1.14229 

.90674 

1.10285 

.93906 

1.06489 

48 

13 

.84556 

1.18264 

.87505 

1.14162 

.90727 

1.10220 

.93961 

1.06427 

47 

14 

.84606 

1.18194 

.87646 

1.14095 

.90781 

1.10156 

.94016 

1.06365 

46 

15 

.84656 

1.18125 

.87698 

1.14028 

.90834 

1.10091 

.94071 

1.06303 

45 

16 

.84706 

1.18055 

.87749 

1.13961 

.90887 

1.10027 

.94125 

1.06241 

44 

17 

.84756 

1.17986 

.87801 

1.13894 

.90940 

1.00963 

.94180 

1.06179 

43 

18 

.84806 

1.17916 

.87852 

1.13828 

.90993 

1.09899 

.94235 

1.06117 

42 

iy 

.84856 

1.17846 

.87904 

1 . 13761 

.91046 

1.09834 

.94200 

1.06056 

41 

20 

.84006 

1.17777 

.87955 

1.13694 

.91099 

1.09770 

.04345 

1.05994 

40 

21 

.84056 

1.17708 

.88007 

1.13627 

.91153 

1.09706 

.94400 

1.05932 

39 

22 

.85006 

1.17638 

.88059 

1.13561 

.91206 

1.09642 

.94455 

1.05870 

38 

23 

.85057 

1.17569 

.88110 

1.13494 

.91259 

1.09578 

.94510 

1.05800 

37 

24 

.85107 

1.17500 

.88162 

1.13428 

.91313 

1.09514 

.94565 

1.05747 

36 

25 

.85157 

1.17430 

.88214 

1.13361 

.91366 

1.09450 

.94620 

1.05685 

35 

26 

.85207 

1.17361 

.88265 

1.13295 

.91419 

1.09386 

.94676 

1.05624 

34 

27 

.85257 

1.17292 

.88317 

1.13228 

.91473 

1.09322 

.94731 

1.05562 

33 

28 

.85308 

1.17223 

.88369 

1.13162 

.91526 

1.09258 

.94786 

1.05501 

32 

20 

.85358 

1.17154 

.88421 

1.13096 

.91580 

1.09195 

.94841 

1.05430 

31 

30 

.85408 

1.17085 

.88473 

1.13029 

.91633 

1.09131 

.94896 

1.05378 

30 

31 

.85458 

1.17016 

.88524 

1.12963 

.91687 

1.09067 

.94952 

1.05317 

29 

32 

.85500 

1.16947 

.88576 

1.12897 

.91740 

1.09003 

.95007 

1.05255 

28 

33 

.85550 

1.16878 

.88628 

1.12831 

.91794 

1.08940 

.95062 

1.05194 

27 

34 

.85600 

1.16809 

.88680 

1.12765 

.91847 

1.08876 

.95118 

1.05133 

26 

35 

.85660 

1.16741 

.88732 

1 . 12699 

.91901 

1.08813 

.95173 

1.05072 

25 

36 

.85710 

1.16672 

.88784 

1.12633 

.91955 

1.08749 

.95229 

1.05010 

24 

37 

.85761 

1.16603 

.88836 

1.12567 

.92008 

1.08686 

.95284 

1.04949 

23 

38 

.85811 

1.16535 

.88888 

1.12501 

.92062 

1.08622 

.95340 

1.04888 

22 

30 

.85862 

1.16466 

.88940 

1.12435 

.92116 

1.08559 

.95395 

1.04827 

21 

40 

.85012 

1.16398 

.88992 

1.12369 

.92170 

1.08496 

.95451 

1.04766 

20 

41 

.85063 

1.16329 

.89045 

1.12303 

.92224 

1.08432 

.95506 

1.04705 

19 

42 

.86014 

1.16261 

.89097 

1.12238 

.92277 

1.08360 

.95562 

1.04644 

18 

43 

.86064 

1.16192 

.89149 

1.12172 

.92331 

1.08306 

.95618 

1.04583 

17 

44 

.86115 

1.16124 

.89201 

1.12106 

.92385 

1.08243 

.95673 

1.04522 

16 

45 

.86166 

1.16056 

.89253 

1.12041 

.92439 

1.08179 

.95729 

1.04461 

15 

46 

.86216 

1.15987 

.89306 

1.11975 

.92493 

1.08116 

.95785 

1.04401 

14 

47 

.86267 

1.15919 

.89358 

1.11909 

.92547 

1.08053 

.95841 

1.04340 

13 

48 

.86318 

1.15851 

.89410 

1.11844 

.92601 

1.07090 

.9.5897 

1.04279 

12 

40 

.86368 

1.15783 

.89463 

1.11778 

.92655 

1.07927 

.95952 

1.04218 

11 

50 

.86410 

1.15715 

.89515 

1.11713 

.92709 

1.07864 

.96008 

1.04158 

10 

51 

.86470 

1.15647 

.89567 

1.11648 

.92763 

1.07801 

.96064 

1.04097 

9 

52 

.86521 

1.15579 

.89620 

1.11582 

.92817 

1.07738 

.96120 

1.04036 

8 

53 

.86572 

1.15511 

.89672 

1.11517 

.92872 

1.07676 

.96176 

1.03976 

7 

54 

.86623 

1.15443 

.89725 

1.11452 

.92926 

1.07613 

.96232 

1.03915 

6 

55 

.86674 

1.15375 

.89777 

1.11387 

.92980 

1.07550 

.96288 

1.03855 

D 

56 

.86725 

1.15308 

.89830 

1.11321 

.93034 

1.07487 

.96344 

1.03794 

4 

57 

.86776 

1.15240 

.80883 

1.11256 

.93088 

1.07425 

.96400 

1.03734 

3 

58 

.86827 

1.15172 

.89035 

1.11191 

.93143 

1.07362 

.96457 

1.03674 

2 

50 

.86878 

1.15104 

.89988 

1.11126 

.93197 

1.07299 

.96513 

1.03613 

1 

60 

.86020 

1.15037 

.90040 

1.11061 

*93252 

1.07237 

.96569 

1.03553 

0 

M. 

Cotang. 

Tang. 

Cotang. 

Tailg. 

Cotang. 

Tang. 

Cotang. 

Tang. 



49° 

48° 

47° 

46° 















































TABLE III. NATURAL TANGENTS, ETC, 


39 



44° 



44° 



44° 


M. 

Tang. 

Cotang. 

M. 

M. 

Tang. 

Cotang. 

M. 

M. 

Tang. 

Cotang. 

M. 

0 

.96569 

1.03553 

60 

20 

.97700 

1.02355 

40 

40 

.98843 

1.01170 

20 

1 

.96625 

1.03493 

59 

21 

.97756 

1.02295 

39 

41 

.98901 

1.01112 

19 

2 

.96681 

1.03433 

58 

22 

.97813 

1.02236 

38 

42 

.98958 

1.01053 

18 

3 

.96738 

1.03372 

57 

23 

.97870 

1.02176 

37 

43 

.99016 

1.00994 

17 

4 

.96794 

1.03312 

56 

24 

.97927 

1.02117 

36 

44 

.99073 

1.00935 

16 

5 

.96850 

1.03252 

55 

25 

.97984 

1.02057 

35 

45 

.99131 

1.00876 

15 

G 

.96907 

1.03192 

54 

26 

.98041 

1.01998 

34 

40 

.99189 

1.00818 

14 

7 

.96963 

1.03132 

53 

27 

.98098 

1.01939 

33 

47 

.99247 

1.00759 

13 

8 

.97020 

1.03072 

52 

28 

.98155 

1.01879 

32 

48 

.99304 

1.00701 

12 

y 

.97076 

1.03012 

51 

29 

.98213 

1.01820 

31 

49 

.99362 

1.00642 

11 

10 

.97133 

1.02952 

50 

30 

.98270 

1.01761 

30 

50 

.99420 

1.00583 

10 

n 

.97189 

1.02892 

49 

31 

.98327 

1.01702 

29 

51 

.99478 

1.00525 

9 

12 

.97246 

1.02832 

48 

32 

.98384 

1.01642 

28 

52 

.99536 

1.00467 

8 

18 

.97302 

1.02772 

47 

33 

.98441 

1.01583 

27 

53 

.99594 

1.00408 

7 

14 

.97359 

1.02713 

46 

34 

.98499 

1.01524 

26 

54 

.99652 

1.00350 

6 

15 

.97416 

1.02653 

45 

35 

.98556 

1.01465 

25 

55 

.99710 

1.00291 

5 

16 

.97472 

1.02593 

44 

36 

.98613 

1.01406 

24 

56 

.99768 

1.00233 

4 

17 

.97529 

1.02533 

43 

37 

.98671 

1.01347 

23 

57 

.99826 

1.00175 

3 

18 

.97586 

1.02474 

42 

38 

.98728 

1.01288 

22 

58 

.99884 

1.00116 

2 

19 

.97643 

1.02414 

41 

39 

.98786 

1.01229 

21 

59 

.99942 

1.00058 

1 

20 

.97700 

1.02355 

40 

40 

.98843 

1.01170 

20 

60 

1.00000 

1.00000 

0 

M. 

Co tang. 

Tang. 

M. 

M. 

Cotang. 

Tang. 

M. 

M. 

Cotang. 

Tang. 

M 


45° 



45 



45° 



TABLE IV. 

— 


LOGARITHMIC SINES, COSINES, 
TANGENTS, 

AND 


COTANGENTS 

























































TABLE IV. LOGARITHMIC SINES, ETC, 


40 


0 ° 179° 


M. 

Sine. 

■ D. 1 

Cosine. 

D. 1* 

Tang. 

D. I”. 

Cotang. 

M. 

0 

1 

2 

3 

4 

5 

6 

7 

8 
y 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 
21 
22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 
67 

58 

59 

60 

Inf. neg. 

6.463726 
.764756 
.940847 

7.065786 
.162696 
.241877 
.308824 
.366816 
.417968 

7.463726 
.505118 
.542906 
.577668 
.609853 
.639816 
.667845 
.694173 
.718997 
.742477 

7.764754 

.785943 

.806146 

.825451 

.843934 

.861662 

.878695 

.895085 

.910879 

.926119 

7.940842 

.955082 

.968870 

.982233 

.995198 

8.007787 

.020021 

.031919 

.043501 

.054781 

8.065776 

.076500 

.086965 

.097183 

.107167 

.116926 

.126471 

.135810 

.144953 

.153907 

8.162681 

.171280 

.179713 

.187985 

.196102 

.204070 

.211895 

.219581 

.227134 

.234557 

.241855 

5017.17 
2934.85 

2082.31 

1615.17 
1319.68 

1115.75 

966.53 

852.54 
762.63 

689.88 

629.81 
579.36 

536.41 

499.38 

467.14 

438.81 

413.72 
391.35 
371.27 

353.15 

336.72 

321.75 
308.05 
295.47 

283.88 

273.17 

263.23 
253.99 

245.38 

237.33 

229.80 

222.73 
216.08 

209.81 
203.90 

198.31 
193.02 
188.01 
183.25 

178.72 

174.41 

170.31 

166.39 

162.65 
159.08 

155.66 
152.38 

149.24 
146.22 

143.33 
140.54 

137.86 
135.29 
132.80 

130.41 
128.10 

125.87 

123.72 
121.64 

0.000000 

.000000 

.000000 

.000000 

.000000 

.000000 

9.999999 

.999999 

.999999 

.999999 

9.999998 

.999998 

.999997 

.999997 

.999996 

.999996 

.999995 

.999995 

9.999994 

.999993 

9.999993 
.999992 
.999991 
.999990 
.999989 
.999988 
.999988 ' 
.999987 
.999986 
.999985 

9.999983 

.999982 

.999981 

.999980 

.999979 

.999977 

.999976 

.999975 

.999973 

.999972 

9.999971 

.999969 

.999968 

.999966 

.999964 

.999963 

.999961 

.999959 

.999958 

.999956 

9.999954 

.999952 

.999950 

.999948 

.999946 

.999944 

.999942 

.999940 

.999938 

.999936 

.999934 

.00 

.00 

.00 

.00 

.00 

.00 

.01 

.01 

.01 

.01 

.01 

.01 

.01 

.01 

.01 

.01 

.01 

.01 

.01 

.01 

.01 

.01 

.01 

.01 

.02 

.02 

.02 

.02 

.02 

.02 

.02 

.02 

.02' 

.02 

.02 

.02 

.02 

.02 

.02 

.02 

.02 

.02 

.02 

.02 

.03 

.03 

.03 

.03 

.03 

‘.03 

.03 

.03 

.03 

.03 

.03 

.03 

.04 

.04 

.01 

.04 

Inf. neg. 

6.463726 
.764756 
.940847 

7.065786 

.162696 

.241878 

.308825 

.366817 

.417970 

7.463727 
.505120 
.542909 
.577672 
.609857 
.639820 
.667849 
.694179 
.719003 
.742484 

7.764761 

.785951 

.806155 

.825460 

.843944 

.861674 

.878708 

.895099 

.910894 

.926134 

7.940858 

.955100 

.968889 

.982253 

.995219 

8.007809 

.020045 

.031945 

.043527 

.054809 

8.065806 
.076531 
.086997 
.097217 
.107202 
.116963 
.126510 
.135851 
.144996 
.153952 

8.162727 
.171328 
.179763 
.188036 
.196156 
.204126 
.211953 
.219641 
.227195 
.234621 
.241921 

5017.17 
2934.83 
2082.31 

1615.17 
1319.69 
1115.78 

996.53 

852.54 
762.63 

689.88 

629.81 

579.33 

536.42 

499.39 
467.15 

438.82 

413.73 

391.36 
371.28 

351.36 

336.73 
321.76 
308.06 
295.49 

283.90 
273.18 
263.25 
254.01 

245.40 

237.35 
229.81 

222.75 
216.10 

209.83 
203.92 

198.33 
193.05 
188.03 

183.27 

178.74 

174.44 

170.34 

166.42 
162.68 
159.10 

155.68 

152.41 

149.27 

146.27 

143.36 
140.57 

137.90 
135.32 

132.84 

130.44 
128.14 

125.90 

123.76 

121.68 

Infinite. 

13.536274 

.235244 

.059153 

12.934214 

.837304 

.758122 

.691175 

.633183 

.582030 

12.536273 

.494880 

.457091 

.422328 

.390143 

.360180 

.332151 

.305821 

.280997 

.257516 

12.235239 

.214049 

.193845 

.174540 

.156056 

.138326 

.121292 

.104901 

.089106 

.073866 

12.059142 

.044900 

.031111 

.017747 

,004781 

11.992191 

.979955 

.968055 

.956473 

.945191 

11.934194 

.923469 

.913003 

.902783 

.892797 

.883037 

.873490 

.864149 

.855004 

.846048 

11.837273 

.828672 

.820237 

.811964 

.803844 

.795874 

.788047 

.780359 

.772805 

.765379 

.758079 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 

M. 

Cosine. 

D. 1”. 

Sine. 

D. 1". 

Cotang. 

D. 1". 

Tang. 

M. 


90 ° 































































1 


TABLE IV. LOGARITHMIC SINES, ETC. 


41 


178° 


M. 

Sine. 

D.l”. 

Cosine. 

D.l" 

Tang. 

D.l ". 

Cotang. 

M. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 
21 
22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 
19 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 

8.241855 

.249033 

.256094 

.263042 

.269881 

.276614 

.283243 

.289773 

.296207 

.302546 

8.308794 

.314954 

.321027 

.327016 

.332924 

.338753 

.344504 

.350181 

.355783 

.361315 

8.366777 

.372171 

.377499 

.382762 

.387962 

.393101 

.398179 

.403199 

.408161 

.413068 

8.417919 

.422717 

.427462 

.432156 

.436800 

.441394 

.445941 

.450440 

.454893 

.459301 

8.463665 

.467985 

.472263 

.476498 

.480693 

.484848 

.488963 

.493040 

.497078 

.501080 

8.505045 

.508974 

.512867 

.516726 

.520551 

.524343 

.528102 

.531828 

535523 

.539186 

.542819 

119.63 

117.68 

115.80 

113.98 

112.21 

110.50 

108.83 

107.21 

105.65 
104.13 

102.66 

101.22 
99.82 

98.47 
97.14 

95.86 

94.60 
93.38 

92.19 
91.03 

89.90 
88.80 

87.72 
86.67 

85.64 

84.64 
83.66 
82.71 

81.77 

80.86 

79.96 

79.09 

78.23 
77.40 

76.57 

75.77 
74.99 
74.22 
73.46 

72.73 

72.00 

71.29 

70.60 

69.91 

69.24 
68.59 
67.94 

67.31 
66.69 
66.08 

65.48 
64.89 

64.31 
63.75 

63.19 

62.64 
62.11 

61.58 
61.06 
60.55 

9.999934 

.999932 

.999929 

.999927 

.999925 

.999922 

.999920 

.999918 

.999915 

.999913 

9.999910 

.999907 

.999905 

.999902 

.999899 

.999897 

.999894 

.999891 

.999888 

.999885 

9.999882 

.999879 

.999876 

.999873 

.999870 

.999867 

.999864 

.999861 

.999858 

.999854 

9.999851 

.999848 

.999844 

.999841 

.999838 

.999834 

.999831 

.999827 

.999823 

.999820 

9.999816 

.999812 

.999809 

.999805 

.999801 

.999797 

.999793 

.999790 

.999786 

.999782 

9.999778 

.999774 

.999769 

.999765 

.999761 

.999757 

.999753 

.999748 

.999744 

.999740 

.999735 

.04 

.04 

.04 

.04 

.04 

.04 

.04 

.04 

.04 

.04 

.04 

.04 

.04 

.04 

.05 

.05 

.05 

.05 

.05 

.05 

.05 

.05 

.05 

.05 

.05 

.05 

.05 

.05 

.05 

.05 

.06 

.06 

.06 

.06 

.06 

.06 

.06 

.06 

.06 

.06 

.06 

.06 

.06 

.06 

.06 

.07 

.07 

.07 

.07 

.07 

.07 

.07 

.07 

.07 

.07 

.07 

.07 

.07 

.07 

.07 

8.241921 

.249102 

.256165 

.263115 

.269956 

.276691 

.283323 

.289856 

.296292 

.302634 

8.308884 

.315046 

.321122 

.327114 

.333025 

.338856 

.344610 

.350289 

.355895 

.361430 

8.366895 

.372292 

.377622 

.382889 

.388092 

.393234 

.398315 

.403338 

.408304 

.413213 

8.418068 

.422869 

.427618 

.432315 

.436962 

.441560 

.446110 

.450613 

.455070 

.459481 

8.463849 

.468172 

.472454 

.476693 

.480892 

.485050 

.489170 

.493250 

.497293 

.501298 

8.505267 

.509200 

.513098 

.516961 

.520790 

.524586 

.528349 

.532080 

.535779 

.539447 

.543084 

119.67 

117.72 

115.84 

114.02 

112.25 
110.54 
108.87 

107.26 

105.70 
104.18 

102.70 

101.26 
99.87 

98.51 
97.19 

95.90 

94.65 
93.43 
92.24 
91.08 

89.95 
88.85 
87.77 
86.72 

85.70 

84.70 

83.71 

82.76 
81.82 

80.91 

80.02 

79.14 

78.30 
77.45 
76.63 
75.83 
75.05 
74.28 

73.52 
72.79 

72.06 

71.35 

70.66 
69.98 

69.31 

68.65 
68.01 

67.38 

66.76 

66.15 

65.55 

64.96 

64.39 
63.82 
63.26 

62.72 
62.18 

61.65 
61.13 
60.62 

11.758079 

.750898 

.743835 

.736885 

.730044 

.723309 

.716677 

.710144 

.703708 

.697366 

11.691116 

.684954 

.678878 

.672886 

.666975 

.661144 

.655390 

.649711 

.644105 

.638570 

11.633105 

.627708 

.622378 

.617111 

.611908 

.606766 

.601685 

.596662 

.591696 

.586787 

11.581932 

.577131 

.572382 

.567685 

.563038 

.558440 

.553890 

.549387 

.544930 

.540519 

11.536151 
.531828 
.527546 
.523307 
.519108 
.514950 
.510830 
.506750 
.502707 
.498702 

11.494733 

.490800 

.486902 

.483039 

.479210 

.475414 

.471651 

.467920 

.464221 

.460553 

.456916 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 

M. 

Cosine. 

D.l . 

Sine. 

D.l ". 

Cotang. 

D.l". 

Tang. 

M. 


91° 


D 


88 








































TABLE IV. LOGARITHMIC SINES, ETC, 


177 


42 


2 ° 


M. 

Sine. 

D.l". 

Cosine. 

D.l'. 

Tang. 

D.l'. 

Cotang. 

M. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 
lfi 

17 

18 

19 

20 
21 
22 

23 

24 

25 
20 

27 

28 

29 

30 

31 

32 

33 

34 
36 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 
40 

47 

48 

49 

50 

51 

52 

53 

54 

55 
50 

57 

58 

59 

60 

8.542819 

.546422 

.549995 

.553539 

.557054 

.560540 

.563999 

.567431 

.570836 

.574214 

8.577506 

.580892 

.584193 

.587469 

.590721 

.593948 

.597152 

.600332 

.603489 

.006623 

8.009734 

.612823 

.015891 

.618937 

.021902 

.624905 

.627948 

.030911 

.033854 

.636770 

8.039080 

.042503 

.045428 

.648274 

.651102 

.053911 

.65G702 

.659475 

.662230 

.004968 

8.007689 

.670393 

.673080 

.075751 

.078405 

.681043 

.683605 

.080272 

.688803 

.691438 

8.093998 

.090543 

.099073 

.701589 

.701090 

.706577 

.709049 

.711507 

.713952 

.716383 

.718800 

60.04 

59.55 

59.06 

58.58 

58.11 
57.65 

57.19 

56.74 
56.30 
55.87 

55.44 

55.02 

54.60 

54.19 

53.79 
53.39 
53.00 

52.61 

52.23 
51.86 

51.49 

51.12 
50.76 

50.41 
50.06 
49.72 

49.38 
49.04 
48.71 

48.39 

48.00 

47.75 

47.43 

47.12 
40.82 
46.52 
40.22 

45.92 
45.03 
45.35 

45.00 

44.79 

44.51 

44.24 

43.97 
43.70 

43.44 
43.18 

42.92 
42.07 

42.42 
42.17 

41.92 
41.68 

41.44 
41.21 

40.97 
40.74 

40.51 
40.29 

9.999735 

.999731 

.999726 

.999722 

.999717 

.999713 

.999708 

.999704 

.999699 

.999694 

9.999089 

.999685 

.999080 

.999675 

.999070 

.999065 

.999000 

.999055 

.999650 

.999645 

9.999040 

.999635 

.999629 

.999624 

.999619 

.999614 

.999608 

.99960.3 

.999597 

.999592 

9.999586 

.999581 

.999575 

.999570 

.999504 

.999558 

.999553 

.999547 

.999541 

.999535 

9.999529 

.999524 

.999518 

.999512 

.999506 

.999500 

.999493 

.999487 

.999481 

.999475 

9.999469 

.999403 

.999450 

.999450 

.999443 

.999437 

.999431 

.999424 

.999418 

.999411 

.999404 

.07 

.07 

.07 

.08 

.08 

.08 

.08 

.08 

.08 

.08 

.08 

.08 

.08 

.08 

.08 

.08 

.08 

.08 

.08 

.09 

.09 

.09 

.09 

.09 

.09 

.09 

.09 

.09 

.09 

.09 

.09 

.09 

.09 

.09 

.09 

.10 

.10 

.10 

.10 

.10 

.10 

.10 

.10 

.10 

.10 

.10 

.10 

.10 

.10 

.10 

.10 

.11 

.11 

.11 

.11 

.11 

.11 

.11 

.11 

.11 

8.543084 

.546691 

.550268 

.553817 

.557336 

.560828 

.564291 

.567727 

.571137 

.574520 

8.577877 

.581208 

.584514 

.587795 

.591051 

.594283 

.597492 

.600677 

.603839 

.606978 

8.610094 

.613189 

.616262 

.619313 

.622343 

.625352 

.628340 

.631308 

.634256 

.637184 

8.640093 
.642982 
.645853 
.648704 
.651537 
.654352 
.657149 
.659928 
.662689 
.665433 

8.668160 
.670870 
.673563 
.676239 
.678900 
.681544 
.684172 
.686784 
.689381 
.691963 

8.694529 

.697081 

.699617 

.702139 

.704646 

.707140 

.709618 

.712083 

.714534 

.716972 

.719396 

60.12 

69.62 

59.14 
58.66 
68.19 

57.73 

57.27 
56.82 
56.38 
55.95 

55.52 
55.10 
54.68 

54.27 

53.87 

53.47 
53.08 
52.70 
52.32 
61.94 

51.58 

51.21 
50.85 
50.50 

50.15 
49.81 

49.47 
49.13 

48.80 

48.48 

48.16 

47.84 

47.53 

47.22 
46.91 

46.61 

46.31 
46.02 

45.73 
45.44 

45.16 

44.88 

44.61 
44.34 
44.07 

43.80 

43.54 

43.28 
43.03 
42.77 

42.52 

42.28 
42.03 
41.79 

41.55 

41.32 
41.08 

40.85 

40.62 
40.40 

11.456916 

.453309 

.449732 

.446183 

.442664 

.439172 

.435709 

.432273 

.428863 

.425480 

11.422123 

.418792 

.415486 

.412205 

.408949 

.405717 

.402508 

.399323 

.396161 

.393022 

11.389906 

.386811 

.383738 

.380687 

.377657 

.374648 

.371660 

.368692 

.365744 

.362816 

11.359907 

.357018 

.354147 

.351296 

.348163 

.345648 

.342851 

.340972 

.337311 

.334.567 

11.331840 

.329130 

.326437 

.323761 

.321100 

.318456 

.315828 

.313216 

.310619 

.308037 

11.30,5471 
.302919 
.300383 
.297861 
.295354 
.292860 
.290382 
.287917 
.285465 
.283u28 
.280604 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 

M. 

Cosine. 

D.l". 

Sine. 

D.l’- 

Cotang. 

D.l”. 

Tang. 

M. 


92 


87 










































3 


TABLE IV. LOGARITHMIC SINES, ETC 


43 


M. 

Sine. 

D. l”. 

Cosine. 

D. 1". 

Tang. 

D.l". 

Cotang. 

M. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 
21 
22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 

8.718800 

.721204 

.723595 

.725972 

.728337 

.730688 

.733027 

.735354 

.737667 

.789969 

8.742259 

.744536 

.746802 

.749055 

.751297 

.753528 

.755747 

.757955 

.760151 

.762337 

8.764511 

.766675 

.768828 

.770970 

.773101 

.775223 

.777333 

.779434 

.781524 

.783605 

8.785675 
.787736 
.789787 
.791828 
.793859 
.795881 
.797894 
.799897 
.801892 
.803876 

8.805852 

.807819 

.809777 

.811726 

.813667 

.815599 

.817522 

.819436 

.821343 

.823240 

8.825130 

.827011 

.828884 

.830749 

.832607 

.834456 

.836297 

.838130 

.839956 

.841774 

.843585 

40.06 

39.84 

39.62 

39.41 
39.19 

38.98 

38.77 
38.57* 

38.36 

38.16 

37.96 

37.76 

37.56 

37.37 

37.17 

36.98 
36.79 
36.61 

36.42 
36.24 

36.06 
35.88 

35.70 

35.53 
35.35 

35.18 
35.01 

34.84 
34.67 
34.51 

34.31 

34.18 
34.02 
33.86 

33.70 

33.54 
33.39 
33.23 
33.08 
32.93 

32.78 

32.63 

32.49 

32.34 

32.19 
32.05 
31.91 

31.77 

31.63 

31.49 

31.35 
31.22 
31.08 
30.95 
30.82 
30.69 

30.56 

30.43 
30.30 
30.17 

9.999404 

.999398 

.999391 

.999384 

.999378 

.999371 

.999364 

.999357 

.999350 

.999343 

9.999336 

.999329 

.999322 

.999315 

.999308 

.999301 

.999294 

.999286 

.999279 

.999272 

9.999265 

.999257 

.999250 

.999242 

.999235 

.999227 

.999220 

.999212 

.999205 

.999197 

9.999189 

.999181 

.999174 

.999166 

.999158 

.999150 

.999142 

.999134 

.999126 

.999118 

9.999110 

.999102 

.999094 

.999086 

.999077 

.999069 

.999061 

.999053 

.999044 

.999036 

9.999027 

.999019 

.999010 

.999002 

.998993 

.998984 

.998976 

.998967 

.998958 

.998950 

.998941 

.11 

.11 

.11 

.11 

.11 

.11 

.12 

.12 

.12 

.12 

.12 

.12 

.12 

.12 

.12 

.12 

.12 

.12 

.12 

.12 

.12 

.12 

.13 

.13 

.13 

.13 

.13 

.13 

.13 

.13 

.13 

.13 

.13 

.13 

.13 

.13 

.13 

.13 

.13 

.13 

.13 

.13 

.14 

.14 

.14 

.14 

.14 

.14 

.14 

.14 

.14 

.14 

.14 

.14 

.14 

.14 

.14 

.15 

.15 

.15 

8.719396 

.721806 

.724204 

.726588 

.728959 

.731317 

.733663 

.735996 

.738317 

.740626 

8.742922 

.745207 

.747479 

.749740 

.751989 

.754227 

.756453 

.758668 

.760872 

.763065 

8.765246 

.767417 

.769578 

.771727 

.773866 

.775995 

.778114 

.780222 

.782320 

.784408 

8.786486 

.788554 

.790613 

.792662 

.794701 

.796731 

.798752 

.800763 

.802765 

.804758 

8.806742 

.808717 

.810683 

.812641 

.814589 

.816529 

.818461 

.820384 

.822298 

.824205 

8.826103 

.827992 

.829874 

.831748 

.833613 

.835471 

.837321 

.839163 

.840998 

.842825 

.844644 

40.17 

39.95 
39.74 

39.52 

39.30 
39.09 
38.89 

38.68 

38.48 
38.27 

38.07 
37 87 

37.68 

37.49 
37.29 

37.10 

36.92 
36.73 
36.55 

36.36 

36.18 
36.00 

35.83 
35.65 
35.48 

35.31 

35.14 
34.97 
34.80 
34.64 

34.47 

34.31 

34.15 
33.99 

33.83 

33.68 

33.52 

33.37 

33.22 
33.07 

32.92 
32.78 

32.62 

32.48 
32.33 

32.19 
32.05 
31.91 
31.77 

31.63 

31.50 
31.36 

31.23 

31.10 

30.96 

30.83 
30.70 
30.57 
30.45 

30.32 

11.280604 

.278194 

.275796 

.273412 

.271041 

.268683 

.266337 

.264004 

.261683 

.259374 

11.257078 

.254793 

.252521 

.250260 

.248011 

.245773 

.243547 

.241332 

.239128 

.236935 

11.234754 

.232583 

.230422 

.228273 

.226134 

.224005 

.221886 

.219778 

.217680 

.215592 

11.213514 
.211446 
.209387 
.207338 
.205299 
.203269 
.201248 
.199237 
.197235 
.195242 

11.193258 
.191283 
.189317 
.187359 
.185411 
.183471 
.181539 
.179616 
.177702 
.175795 

11.173897 
.172008 
.170126 
.168252 
.166387 
.164529 
.162679 
.160837 
.159002 
.157175 
.155356 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

32 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

0 

8 

7 

6 

5 

4 

3 

2 

1 

0 

M. 

Cosine. 

D. 1". 

Sine. 

D.l". 

Cotang. 

D. 1 

Tailg. 

M. 


93 


86 ° 





























44 

4* 


TABLE IV. LOGARITHMIC SINES, ETC, 


175 


M. 

Sine. 

D. 1”. 

Cosine. 

D. 1". 

Tang. 

D. 1”. 

Cotang. 

M. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 
21 
22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 
43 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 

8.843585 

.845387 

.847183 

.848971 

.850751 

.852525 

.854291 

.856049 

.857801 

.859546 

8.861283 

.863014 

.864738 

.866455 

.868165 

.869868 

.871565 

.873255 

.874938 

.876615 

8.878285 

.879949 

.881607 

.883258 

.884903 

.886542 

.888174 

.889801 

.891421 

.893035 

8.894043 

.896246 

.897842 

.899432 

.901017 

.902596 

.904169 

.905736 

.907297 

.908853 

8.910404 

.911949 

.913488 

.915022 

.916550 

.918073 

.919591 

.921103 

.922610 

.924112 

8.925609 

.927100 

.928587 

.930068 

.931544 

.933015 

.934481 

.935942 

.937398 

.938850 

.940296 

30.05 

29.92 

29.80 
29.67 

29.55 
29.43 

29.31 

29.19 
29.07 
28.96 

28.84 

28.73 
28.61 

28.50 
28.39 
28.28 
28.17 
28.06 
27.95 

27.86 

27.73 
27.63 
27.52 

27.42 

27.31 

27.21 

27.11 
27.00 
26.90 

26.80 

26.70 

26.60 

26.51 
26.41 

26.31 

26.22 

26.12 
26.03 

25.93 

25.84 

25.75 

25.66 

25.56 
25.47 
25.38 
25.29 

25.20 
25.12 
25.03 

24.94 

24.86 
24.77 
24.69 
24.60 

24.52 

24.43 
24.35 
24.27 
24.19 
24.11 

9.998941 

.998932 

.998923 

.998914 

.998905 

.998896 

.998887 

.998878 

.998869 

.998860 

9.998851 

.998841 

.998832 

.998823 

.998813 

.998804 

.998795 

.998785 

.998776 

.998766 

9.998757 

.998747 

.998738 

.998728 

.998718 

.998708 

.998690 

.998689 

.998679 

.998669 

9.998659 

.998649 

.998639 

.998629 

.998619 

.998609 

.998599 

.998589 

.998578 

.998568 

9.998558 

.998548 

.998537 

.998527 

.998516 

.998.506 

.998495 

.998485 

.998474 

.998464 

9.998453 

.998442 

.998431 

.998121 

.908410 

.998399 

.998388 

.998377 

.998366 

.998355 

.998344 

.15 

.15 

.15 

.15 

.15 

.15 

.15 

.15 

.15 

.15 

.15 

.15 

.15 

.16 

.16 

.16 

.16 

.16 

.16 

.16 

.16 

.16 

.16 

.16 

.16 

.16 

.16 

.16 

.16 

.17 

.17 

.17 

.17 

.17 

.17 

.17 

.17 

.17 

.17 

.17 

.17 

.17 

.17 

.17 

.18 

.18 

.18 

.18 

.18 

.18 

.18 

.18 

.18 

.18 

.18 

.18 

.18 

.18 

.18 

.18 

8.844644 

.846455 

.848260 

.850057 

.851846 

.853628 

.855403 

.857171 

.858932 

.860686 

8.862433 

.864173 

.865906 

.867632 

.869351 

.871064 

.872770 

.874469 

.876162 

.877849 

8.879529 

.881202 

.882869 

.884530 

.886185 

.887833 

.889476 

.891112 

.892742 

.894366 

8.895984 

.897596 

.899203 

.900803 

.902398 

.903987 

.90.5570 

.907147 

.908719 

.910285 

8.911846 

.913401 

.914951 

.916495 

.918034 

.919568 

.921096 

.922619 

.924136 

.925049 

8.9271,56 

.928658 

.930155 

.931647 

.933134 

.934616 

.936093 

.937565 

.939032 

.940494 

.941952 

30.19 
30.07 

29.95 

29.82 

29.70 

29.58 

29.46 
29.35 
29.23 

29.11 

29.00 

28.88 

28.77 
28.66 
28.54 
28.43 
28.32 
28.21 

28.11 
28.00 

27.89 

27.79 

27.68 

27.58 

27.47 

27.37 
27.27 
27.17 
27.07 
26.97 

26.87 

26.77 
26.67 

26.58 

26.48 

26.38 

26.29 

26.20 
26.10 
26.01 

25.92 

25.83 
25.74 
25.65 
26.56 
25.47 

25.38 

25.30 
25.21 
25.12 

25.03 

24.95 
24.86 

24.78 

24.70 
24.61 
24.53 
24.45 
24.37 

24.30 

11.155356 

.153545 

.151740 

.149943 

.148154 

.146372 

.144597 

.142829 

.141068 

.139314 

11.137567 
.135827 
.134094 
.132368 
.130649 
.128936 
.127230 
.125531 
.123838 
..122151 

11.120471 

.118798 

.117131 

.115470 

.113815 

.112167 

.110524 

.108888 

.107258 

.105634 

11.104016 

.102404 

.100797 

.099197 

.097602 

.096013 

.094430 

.092853 

.091281 

.089715 

11.088154 

.086599 

.085049 

.083605 

..081966 

.080432 

.078904 

.077381 

.075864 

.074351 

11.072844 

.071342 

.069845 

.068353 

.066866 

.065384 

.063907 

.062435 

.060968 

.059506 

.058048 

60 

59 

58 

67 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

25 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

2 

2 

1 

0 

M. 

Cosine. 

D. 1". 1 Sine. D. 1". 

Cotang. 

1 D. 1 

Tang. 

M. 


94 


85 












































5 


TABLE IV. LOGARITHMIC BINES, ETC, 


45 


174° 


M. 

Sine. 

D.l". 

Cosine. 

D.l". 

Tang. 

D.l". 

Cotang. 

M. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 
21 
22 

23 

24 
26 
26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 
69 

8.940296 

.941738 

.943174 

.944606 

.946034 

.947456 

.948874 

.950287 

.951696 

.953100 

8.954499 

.955894 

.957284 

.958670 

.960052 

.961429 

.962801 

.964170 

.965534 

.966893 

8.968249 

.969600 

.970947 

.972289 

.973628 

.974962 

.976293 

.977619 

.978941 

.980259 

8.981573 

.982883 

.984189 

.985491 

.986789 

.988083 

.989374 

.990660 

.991943 

.993222 

8.994497 

.995768 

.997036 

.998299 

.999560* 

9.000816 

.002069 

.003318 

.004563 

.005805 

9.007044 

.008278 

.009510 

.010737 

.011962 

.013182 

.014400 

.015613 

.016824 

.018031 

.019235 

24.03 

23.94 

23.87 

23.79 
23.71 

23.63 
23.55 
23.48 

23.40 
23.32 

23.25 

23.17 

23.10 
23.02 

22.95 

22.88 

22.80 
22.73 
22.66 
22.59 

22.52 

22.45 

22.38 

22.31 

22.24 

22.17 

22.10 
22.03 
21.97 
21.90 

21.83 

21.77 

21.70 

21.63 

21.57 
21.50 
21.44 

21.38 

21.31 

21.25 

21.19 

21.12 

21.06 

21.00 

20.94 

20.88 

20.82 

20.76 

20.70 

20.64 

20.58 

20.52 

20.46 

20.40 
20.34 
20.29 
20.23 

20.17 
20.12 
20.06 

9.998344 

.998333 

.998322 

.998311 

.998300 

.998289 

.998277 

.998266 

.998255 

.998243 

9.998232 

.998220 

.998209 

.998197 

.998186 

.998174 

.998163 

.998151 

.998139 

.998128 

9.998116 

.998104 

.998092 

.998080 

.998068 

.998056 

.998044 

.998032 

.998020 

.998008 

9.997996 

.997984 

.997972 

.997959 

.997947 

.997935 

.997922 

.997910 

.997897 

.997885 

9.997872 

.997860 

.997847 

.997835 

.997822 

.997809 

.997797 

.997784 

.997771 

, .997758 

9.997745 

.997732 

.997719 

.997706 

.997693 

.997680 

.997667 

.997654 

.997641 

.997628 

.997614 

.19 

.19 

.19 

.19 

.19 

.19 

.19 

.19 

.19 

.19 

' .19 
.19 
.19 
.19 
.19 
.19 
.19 
.19 
.20 
.20 

.20 

.20 

.20 

.20 

.20 

.20 

.20 

.20 

.20 

.20 

.20 

.20 

.20 

.20 

.20 

.21 

.21 

.21 

.21 

.21 

.21 

.21 

.21 

.21 

.21 

.21 

.21 

.21 

.21 

.21 

.21 

.21 

.21 

.21 

.22 

.22 

.22 

.22 

.22 

.22 

8.941952 

.943404 

.944852 

.946295 

.947734 

.949168 

.950597 

.952021 

.953441 

.954856 

8.956267 

.957674 

.959075 

.960473 

.961866 

.963255 

.964639 

.966019 

.967394 

.968766 

8.970133 

.971496 

.972855 

.974209 

.975560 

.976906 

.978248 

. .979586 
.980921 
.982251 

8.983577 

.984899 

.986217 

.987532 

.988842 

.990149 

.991451 

.992750 

.994045 

.995337 

8.996624 

.997908 

.999188 

9.000465 

.001738 

.003007 

.004272 

.005534 

.006792 

.008047 

9.009298 

.010546 

.011790 

.013031 

.014268 

.015502 

.016732 

.017959 

.019183 

.020403 

.021620 

24.21 

24.13 
24.05 

23.97 

23.90 
23.82 

23.74 

23.67 
23.60 

23.51 

23.44 

23.37 

23.29 

23.22 

23.14 
23.07 
23.00 
22.93 
22.86 

22.79 

22.72 

22.65 

22.57 

22.51 

22.44 

22.37 

22.30 

22.23 
22.17 
22.10 

22.04 

21.97 

21.91 

21.84 
21.78 
21.71 

21.65 

21.58 

21.52 
21.46 

21.40 
21.34 

21.27 
21.21 

21.15 
21.09 
21.03 

20.97 

20.91 

20.85 

20.80 

20.74 

20.68 
20.62 
20.56 
20.51 

20.45 

20.40 
20.33 

20.28 

11.058048 

.056596 

.055148 

.053705 

.052266 

.050832 

.049403 

.047979 

.046559 

.045144 

11.043733 

.042326 

.040925 

.039537 

.038134 

.036745 

.035361 

.033981 

.032606 

.031234 

11.029867 

.028504 

.027145 

.025791 

.024440 

.023094 

.021752 

.020414 

.019079 

.017749 

11.016123 

.015101 

.013783 

.012468 

.011158 

.009851 

.008549 

.007250 

.005955 

.004663 

11.003376 

.002092 

.000812 

10.999535 

.998262 

.996993 

.995728 

.994466 

.993208 

.991953 

10.990702 

.989454 

.988210 

.986969 

.985732 

.984498 

.983268 

.982041 

.980817 

.979597 

.978380 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 

M. 

Cosine, 

D.l". 

Sine. 

D.l”. 

Cotang. 

D.l". 

Tang. 

M. 


95 


84 







































TABLE IV. LOGARITHMIC SINES, ETC, 


46 


M. 

Sine. 

D.l”. 

Cosine. 

D.l”. 

Tang. 

D.l”. 

Cotang. 

M. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

0 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 
21 
22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 
'41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 

9.019235 

.020435 

.021632 

.022825 

.024016 

.025203 

.026386 

.027567 

.028744 

.029918 

9.031089 

.032257 

.033421 

.034582 

.035741 

.036896 

.038048 

.039197 

.040342 

.041485 

9.042625 

.043762 

.044895 

.046026 

.047154 

.048279 

.049400 

.050519 

.051635 

.052749 

9.053859 

.054966 

.056071 

.057172 

.058271 

.059367 

.060460 

.061551 

.062639 

.063724 

9.064806 

.065885 

.066962 

.068036 

.069107 

.070176 

.071242 

.072306 

.073366 

.074424 

9.075480 

.076533 

.077583 

.078631 

.079676 

.080719 

.081759 

.082797 

.083832 

.084864 

.085894 

20.00 

19.95 

19.89 

19.84 
19.78 
19.73 

19.67 
19.62 
19.57 
19.52 

19.47 

19.41 

19.36 

19.30 

19.25 

19.20 
19.15 
19.10 
19.05 
19.00 

18.95 

18.90 

18.85 
18.80 
18.75 
18.70 
18.65 
18.60 

18.55 

18.50 

18.46 

18.41 

18.36 

18.31 
18.27 
18.22 

18.17 
18.13 
18.08 
18.04 

17.99 

17.95 

17.90 

17.86 
17.81 
17.77 
17.72 

17.68 
17.64 
17.59 

17.55 

17.51 

17.46 

17.42 
17.38 
17.34 
17.29 

17.25 

17.21 

17.17 

9.997614 

.997601 

.997588 

.997574 

.997561 

.997547 

.997534 

.997520 

.997507 

.997493 

9.997480 
.997466 
.997452 
.997439 
.997425 
.997411 
.997397 
.997383 
.997369 
.997355 

9.997311 

.997327 

.997313 

.997299 

.997285 

.997271 

.997257 

.997242 

.997228 

.997214 

9.997199 

.997185 

.997170 

.997156 

.997141 

.997127 

.997112 

.997098 

.997083 

.997068 

9997053 

.997039 

.997024 

.997009 

.996994 

.996979 

.996964 

.996949 

.996934 

.996919 

9.996904 
.996889 
.996874 
.996858 
.996843 
.996828 
.996812 
.996797 
.996782 
.99(5766 
.996751 

.22 

.22 

.22 

.22 

.22 

.22 

.23 

.23 

.23 

.23 

.23 

.23 

.23 

.23 

.23 

.23 

.23 

.23 

.23 

.23 

.23 

.24 

.24 

.24 

.24 

.24 

.24 

.24 

.24 

.24 

.24 

.24 

.24 

.24 

.24 

.24 

.24 

.24 

.25 

.25 

.25 

.25 

.25 

.25 

.25 

.25 

.25 

.25 

.25 

.25 

.25 

.25 

.25 

.25 

.25 

.25 

.26 

.26 

.26 

.26 

9.021620 

.022834 

.024044 

.025251 

.026455 

.027655 

.028852 

.030046 

.031237 

.032425 

9.033609 

.034791 

.035969 

.037144 

.038316 

.039485 

.040651 

.041813 

.042973 

.044130 

9.045284 

.046434 

.047582 

.048727 

.049869 

.051008 

.052144 

.053277 

.054407 

.055535 

9.056659 

.057781 

.058900 

.060016 

.061130 

.062240 

.063348 

.064453 

.065556 

.066655 

9.067752 

.068846 

.069938 

.071027 

.072113 

.073197 

.074278 

.075356 

.076432 

.077505 

9.078576 

.079644 

.080710 

.081773 

.082833 

.083891 

.084947 

.086000 

.087050 

.088098 

.089144 

20.23 

20.17 
20.12 
20.06 
20.01 
19.95 
19.90 
19.85 

19.79 

19.74 

19.69 

19.64 

19.58 
19.53 
19.48 
19.43 
19.38 

19.33 

19.28 

19.23 

19.18 
19.13 
19.08 
19.03 
18.98 

18.93 

18.89 
18.84 

18.79 

18.74 

18.70 

18.65 
18.60 
18.56 

18.51 

18.46 

18.42 
18.37 

18.33 

18.28 

18.24 

18.19 
18.15 
18.10 
18.06 
18.02 
17.97 

17.93 

17.89 
' 17.84 

17.80 
17.76 
17.72 
17.67 
17.63 

17.59 
17.55 

17.51 

17.47 

17.43 

10.978380 

.977166 

.975956 

.974749 

.973545 

.972345 

.971148 

.969954 

.968763 

.967575 

10.966391 

.965209 

.964031 

.962856 

.961684 

.960515 

.959349 

.958187 

.957027 

.955870 

10.954716 

.953566 

.952418 

.951273 

.950131 

.948992 

.947856 

.946723 

.945593 

.944465 

10.943341 

.942219 

.941100 

.939984 

.938870 

.937760 

.936652 

.935547 

.934444 

.933345 

10.932248 

.931154 

.930062 

.928973 

.927887 

.926803 

.925722 

.924644 

.923568 

.922495 

10.921424 

.920356 

.919290 

.918227 

.917167 

.916109 

.915053 

.914000 

.912950 

.911902 

.910856 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 

M. 

Cosine. 

D.l”. 

Sine. 

D.l”. 

Cotang. 

1 D.l”. 

Tang. 

M. 


96 ° 

































TABLE IY. LOGARITHMIC SINES, ETC 


47 


172° 


M. 

Sine. 

D. 1". 

Cosine. 

D. 1" 

Tang. 

D. 1". 

| Cotang. 

| M. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 
21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 
40 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 

9.085894 

.086922 

.087947 

.088970 

.089990 

.091008 

.092024 

.093037 

.094047 

.095056 

9.096062 
.097005 
.098066 
.099065 
.100062 
.101056 
.10204S 
.103037 
.104025 
.105010 

9.105992 
.106973 
.107951 
.108927 
.109901 
.110873 
.111842 
.112809 
.113774 
.114737 

9.115698 

.116656 

.117613 

.118567 

.119519 

.120169 

.121417 

.122362 

.123306 

.124248 

9.125187 

.126125 

.127060 

.127993 

.128925 

.129854 

.130781 

.131706 

.132630 

.133551 

9.134470 
.135387 
.136303 
.137216 
.138128 
.139037 
.139944 
.140850 
.141754 
.142655 
.143555 

17.13 
17.09 
17.05 
17.00 
16.96 
16.92 
16.88 
16.84 
16.80 

16.76 

16.73 
16 68 

16.65 
16.61 
16.57 
16.53 

16.49 
16.46 
16.43 

16.38 

16.34 
16.30 
16.27 
16.23 

16.19 
16.16 
16.12 
16.08 
16.05 
16.01 

15.98 

15.94 

15.90 

15.87 

15.83 

15.80 

15.76 

15.73 
15.69 

15.66 

15.62 

15.59 

15.56 

15.52 

15.49 
15.45 
15.42 

15.39 

15.35 
15.32 

15.29 

15.26 

15.22 

15.19 
15.16 

15.13 
15.09 
15.06 
15.03 
15.00 

9.996751 

.996735 

.996720 

.996704 

.996688 

.996673 

.996657 

.996641 

.996625 

.996610 

9.996594 

.996578 

.996562 

.996546 

.996530 

.996514 

.996498 

.996482 

.996465 

.996449 

9.996433 

.996417 

.996400 

.996384 

.996368 

.996351 

.996335 

.996318 

.996302 

.996285 

9.996269 

.996252 

.996235 

.996219 

.996202 

.996185 

.996168 

.996151 

.996134 

.996117 

9.996100 

.996083 

.996066 

.996049 

.996032 

.996015 

.995998 

.995980 

.995963 

.995946 

9.995928 

.995911 

.995894 

.995876 

.995859 

.995841 

.995823 

.995806 

.995788 

.995771 

.995753 

.26 

.26 

.26 

.26 

.26 

.26 

.26 

.26 

.26 

.26 

.26 

.27 

.27 

.27 

.27 

.27 

•27 

.27 

.27 

.27 

.27 

.27 

.27 

.27 

.27 

.27 

.27 

.27 

.28 

.28 

.28 

.28 

.28 

.28 

.28 

.28 

.28 

.28 

.28 

.28 

.29 

.29 

.29 

.29 

.29 

.29 

.29 

.29 

.29 

.29 

.29 

.29 

.29 

.29 

.29 

.29 

.29 

.29 

.29 

.29 

9.089144 

.090187 

.091228 

.092266 

.093302 

.094336 

.095367 

.096395 

.097422 

.098446 

9.099468 

.100487 

.101504 

.102519 

.103532 

.104542 

.105550 

.106556 

.107559 

.108560 

9.109559 

.110556 

.111551 

.112543 

.113533 

.114521 

.115507 

.116491 

.117472 

.118452 

9.119429 

.120404 

.121377 

.122348 

.123317 

.124284 

.125249 

.126211 

.127172 

.128130 

9.129087 

.130041 

.130994 

.131944 

.132893 

.133839 

.134784 

.135726 

.136667 

.137605 

9.138542 
.139476 
.140409 
.141340 
.142269 
.143196 
.144121 
.145044 
.145966 
.146885 
.147803 

17.39 

17.35 

17.31 
17.27 
17.23 
17.19 

17.15 

17.11 
17.07 
17.03 

16.99 

16.95 

16.91 
16.88 

16.84 
16.80 

16.76 
16.72 
16.69 
16.65 

16.61 

16.58 

16.54 

16.50 

16.47 
16.43 

16.39 

16.36 

16.32 

16.29 

16.25 

16.22 

16.18 

16.15 

16.11 
16.08 
16.04 
16.01 
15.98 
15.94 

15.91 
15.87 

15.84 
15.81 

15.77 
15.74 
15.71 
15.68 
15.64 
15.61 

15.58 

15.55 

15.51 

15.48 
15.45 
15.42 

15.39 

15.36 

15.32 

15.29 

10.910856 

.909813 

.908772 

.907734 

.906698 

.905664 

.904633 

.903605 

.902578 

.901554 

10.900532 

.899513 

.898496 

.897481 

.896468 

.895458 

.894450 

.893444 

.892441 

.891440 

10.890441 

.889444 

.888449 

.887457 

.886467 

.885479 

.884493 

.883509 

.882528 

.881548 

10.880571 

.879596 

.878623 

.877652 

.876683 

.875716 

.874751 

.873789 

.872828 

.871870 

10.870913 
.869959 
.869006 
.868056 
.867107 
.866161 
.865216 1 
.864274 ' 
.863333 
.862395 

10.861458 

.860524 

.859591 

.858660 

.857731 

.856804 

.855879 

.854956 

.854034 

.853115 

.852197 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 

M. 

Cosine. 

D. 1”. 

Sine. 

D. 1 '. 

Cotang. 

D. 1 

Tang. 

M. 


97® 82 ° 







































TABLE IV. LOGARITHMIC SINES, ETC, 


171 


48 


8 ° 


M. 

Sine. 

B. 1". 

Cosine. 

D.l'. 

Tang. 

D. 1”. 

Cotang. 

M. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

It 

12 

13 

14 

15 

16 

17 

18 

19 

20 
21 
22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 

9.143555 
.144453 
.14.5349 
.146243 
.147136 
.148026 
.148915 
.149802 
.150686 
.151569 

9.152451 
.153330 
.154208 
.155083 
.155957 
.156830 
.157700 
.158569 
.159435 
.160301 

9.161164 
.162025 
.162885 
.163743 
.164600 
.165454 
.166307 
.167159 
.168008 
.168856 

9.169702 
.170547 
.171389 
.172230 
. 173070 
.173908 
.174744 
.175578 
.176411 
.177242 

9.178072 
.178900 
.179726 
.180551 
.181374 
.182196 
.183016 
.183834 
.184651 
.185466 

9.186280 

.187092 

.187903 

.188712 

.189519 

.190325 

.191130 

.191933 

.192734 

.193534 

.194332 

14.97 

14.93 

14.90 

14.87 

14.84 
14.81 
14.78 

14.75 

14.72 

14.69 

14.66 

14.63 

14.60 
14.57 

14.54 

14.51 

14.48 

14.45 

14.42 
14.39 

14.36 

14.33 

14.30 
14.27 
14.24 
14.22 
14.19 
14.16 
14.13 
14.10 

14.07 
14.05 
14.02 
13.99 
13.96 

13.94 

13.91 

13.88 

13.85 
13.83 

13.80 

13.77 

13.75 

13.72 

13.69 

13.67 

13.64 

13.61 
13.59 
13.56 

13.54 

13.51 

13.48 

13.46 

13.43 
13.41 
13.38 

13.36 

13.33 

13.31 

9.995753 

.995735 

.995717 

.995699 

.995681 

.995664 

.995646 

.995628 

.995610 

.995591 

9.995573 

.995555 

.995537 

.995519 

.995501 

.995482 

.995464 

.995446 

.995427 

.995409 

9.995390 

.995372 

.995353 

.995334 

.995316 

.995297 

.995278 

.995260 

.995241 

.995222 

9.995203 

.995184 

.995165 

.995146 

.995127 

.995108 

.995089 

.995070 

.995051 

.995032 

9.995013 

.994993 

.994974 

.994955 

.994935 

.994916 

.994896 

.994877 

.994857 

.994838 

9.994818 

.994798 

.994779 

.994759 

.994739 

.994719 

.994700 

.994680 

.994660 

.994640 

.994620 

.30 

.30 

.30 

.30 

.30 

.30 

.30 

.30 

.30 

.30 

.30 

.30 

.30 

.30 

.31 

.31 

.31 

.31 

.31 

.31 

.31 

.31 

.31 

.31 

.31 

.31 

.31 

.31 

.32 

.32 

.32 

.32 

.32 

.32 

.32 

.32 

.32 

.32 

.32 

.32 

.32 

.32 

.32 

.32 

.32 

.33 

.33 

.33 

.33 

.33 

.33 

.33 

.33 

.33 

.33 

.&3 

.33 

.33 

.33 

.33 

9.147803 
.148718 
.149632 
.150544 
.151454 
.152363 
.153269 
.154174 
.155077 
.155978 

9.156877 
.157775 
.158671 
.159565 
.160457 
.161347 
.162236 
.163123 
.164008 
.164892 

9.165774 

.166654 

.167532 

.168409 

.169284 

.170157 

.171029 

.171899 

.172767 

.173634 

9.174499 
.175362 
.176224 
.177084 
.177942 
.178799 
.179655 
.180508 
.181360 
.182211 

9.183059 

.183907 

.184752 

.185597 

.186439 

.187280 

.188120 

.188958 

.189794 

.190629 

9.191462 

.192294 

.193124 

.193953 

.194780 

.195606 

.196430 

.197253 

.198074 

.198894 

.199713 

15.26 

15.23 

15.20 

15.17 

15.14 

15.11 
15.08 
15.05 
15.02 

14.99 

14.96 

14.93 

14.90 
14.87 

14.84 

14.81 

14.78 

14.75 

14.73 

14.70 

14.67 

14.64 
14.61 
14.58 
14.56 
14.53 
14.50 
14.47 
14.44 
14.42 

14.39 

14.36 

14:33 

14.31 

14.28 

14.25 

14.23 

14.20 

14.17 

14.15 

14.12 
14.09 
14.07 
14.04 
14.02 

13.99 

13.97 

13.94 

13.91 
13.89 

13.86 

13.84 

13.81 

13.79 

13.76 

13.74 

13.71 
13.69 
13.66 

13.64 

10.852197 

.851282 

.850368 

.849456 

.848546 

.847637 

.846731 

.845826 

.844923 

.844022 

10.843123 

.842225 

.841329 

.840435 

.839543 

.838653 

.837764 

.836877 

.835992 

.835108 

10.834226 

.833346 

.832468 

.831591 

.830716 

.829843 

.828971 

.828101 

.827233 

.826366 

10.825501 

.824638 

.823776 

.822916 

.822058 

.821201 

.820345 

.819492 

.818640 

.817789 

10.816941 

.816093 

.815248 

.814403 

.813561 

.812720 

.811880 

.811042 

.810206 

.809371 

10.808538 

.807706 

.806876 

.806047 

.805220 

.801394 

.803570 

.802747 

.801926 

.801106 

.800287 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 

M. 

Cosine. 

D. 1". 

Sine. 

D.l'. 

Cotang. 

D. 1". 

Tang. 

M. 


98 " 81 o 









































9 


TABLE IY. LOGARITHMIC SINES, ETC, 


49 


170* 


M. 

Sine. 

D. 1”. 

Cosine. 

D.l ". 

Tang. 

D. 1 '. 

Cotang. 

M* 

0 

1 

2 

3 

4 

5 

G 

7 

8 

9 

10 

11 

12 

13 

14 

15 
1G 

17 

18 

19 

20 
21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 
3G 

37 

38 

39 

40 

41 

42 

43 

44 

45 
4G 

47 

48 

49 

50 

51 

52 

53 

54 

55 
5G 

57 

58 

59 

60 

9.194332 

.195129 

.195925 

.196719 

.197511 

.198302 

.199091 

.199879 

.200666 

.201451 

9.202234 

.203017 

.203797 

.204577 

.205354 

.206131 

.206906 

.207679 

.208452 

.209222 

9.209992 

.210760 

.211526 

.212291 

.213055 

.213818 

.214579 

.215338 

.216097 

.216854 

9.217609 

.218363 

.219116 

.219868 

.220618 

.221367 

.222115 

.222861 

.223606 

.224349 

9.225092 

.225833 

.226573 

.227311 

.228048 

.228784 

.229518 

.230252 

.230984 

.231715 

9.232444 

.233172 

.233899 

.234625 

.235349 

.236073 

.236795 

.237515 

.238235 

.238953 

.239670 

13.28 

13.26 

13.23 

13.21 

13.18 

13.16 

13.13 

13.11 
13.08 
13.06 

13.04 

13.01 

12.99 

12.96 

12.94 
12.92 
12.89 
12.87 
12.85 
12.82 

12.80 

12.78 

12.75 

12.73 

12.71 

12.68 

12.66 

12.64 

12.62 

12.59 

12.57 

12.55 

12.53 

12.50 

12.48 

12.46 

12.44 

12.42 

12.39 

12.37 

12.35 

12.33 

12.31 

12.29 

12.26 

12.24 

12.22 
12.20 

12.18 

12.16 

12.14 

12.12 
12.10 
12.07 
12.05 
12.03 
12.01 

11.99 

11.97 

11.95 

9.994620 

.994600 

.994580 

.994560 

.994540 

.994519 

.994499 

.994479 

.994459 

.994438 

9.994418 

.994398 

.994377 

.994357 

.994336 

.994316 

.994295 

.994274 

.994254 

.994233 

9.994212 

.994191 

.994171 

.994150 

.994129 

.994108 

.994087 

.994066 

.994045 

.994024 

9.994003 

.993982 

.993960 

.993939 

.993918 

.993897 

.993875 

.993854 

.993832 

.993811 

9.993789 

.993768 

.993746 

.993725 

.993703 

.993681 

.993660 

.993638 

.993616 

.993594 

9.993572 

.993550 

.993528 

.993506 

.993484 

.993462 

.993440 

.993418 

.993396 

.993374 

.993351 

.33 

.33 

.33 

.34 

.34 

.34 

.34 

.34 

.34 

.34 

.34 

.34 

.34 

.34 

.34 

.34 

.34 

.35 

.35 

.35 

.35 

.35 

.35 

.35 

.35 

.35 

.35 

.35 

.35 

.35 

.35 

.35 

.35 

.35 

.36 

.36 

.36 

.36 

.36 

.36 

.36 

.36 

.36 

.36 

.36 

.36 

.36 

.36 

.36 

.36 

.36 

.37 

.37 

.37 

.37 

.37 

.37 

.37 

.37 

.37 

9.199713 

.200529 

.201345 

.202159 

.202971 

.203782 

.204592 

.205400 

.206207 

.207013 

9.207817 

.208619 

.209420 

.210220 

.211018 

.211815 

.212611 

.213405 

.214198 

.214989 

9.215780 

.216568 

.217356 

.218142 

.218926 

.219710 

.220492 

.221272 

.222052 

.222830 

9.223607 

.224382 

.225156 

.225929 

.226700 

.227471 

.228239 

.229007 

.229773 

.230539 

9.231302 

.232065 

.232826 

.233586 

.234345 

.235103 

.235859 

.236614 

.237368 

.238120 

9.238872 

.239622 

.240371 

.241118 

.241865 

.242610 

.243354 

.244097 

.244839 

.245579 

.246319 

13.62 

13.59 

13.57 

13.54 

13.52 

13.49 

13.47 

13.45 

13.42 

13.40 

13.38 

13.35 

13.33 

13.31 
13.28 
13.26 
13.24 
13.21 
13.19 
13.17 

13.15 

13.12 

13.10 

13.08 

13.06 

13.03 

13.01 

12.99 

12.97 

12.95 

12.92 

12.90 

12.88 

12.86 

12.84 

12.82 

12.79 

12.77 

12.75 

12.73 

12.71 

12.69 

12.67 

12.65 

12.63 

12.60 

12.58 
12.56 

12.54 

12.52 

12.50 

12.48 

12.46 
12.44 

12.42 

12.40 

12.38 

12.36 

12.34 

12.32 

10.800287 

.799471 

.798655 

.797841 

.797029 

.796218 

.795408 

,794600 

.793793 

.792987 

10.792183 

.791381 

.790580 

.789780 

.788982 

.788185 

.787389 

.786595 

.785802 

.785011 

10.784220 

.783432 

.782644 

.781858 

.781074 

.780290 

.779508 

.778728 

.777948 

.777170 

10.776393 

.775618 

.774844 

.774071 

.773300 

.772529 

.771761 

.770993 

.770227 

.769461 

10.768698 

.767935 

.767174 

.766414 

.765655 

.764897 

.764141 

.763386 

.762632 

.761880 

10.761128 

.760378 

.759629 

.758882 

.758135 

.757390 

.756646 

.755903 

.755161 

.754421 

.753681 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

.24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 

M 

Cosine. 

D. 1". 

Sine. 

D. 1”. 

Cotang. 

D. 1”. 

Tang. 

M 


80 ° 






























TABLE IV. LOGARITHMIC SINES, ETC. 


169 


50 


10 ° 


M. 

Sine. 

D. 1". 

Cosine. 

D. 1". 

Tang. 

D. 1". 

Cotang. 

M. 

0 

1 

2 

3 

4 

5 

6 

7 

8 
y 

10 

n 

12 

13 

14 

15 
10 

17 

18 

19 

20 
21 
22 

23 

24 

25 
2G 

27 

28 

29 

30 

31 

32 

33 

34 

35 
3G 

37 

38 

39 

40 

41 

42 

43 

44 

45 
4G 

47 

48 

49 

50 

51 

52 

53 

54 

55 
5G 

57 

58 

59 

60 

9.239G70 
.240380 
.241101 
.241814 
.24252G 
.243237 
.243947 
.244G5G 
.245303 
.24G0G9 

9.246775 
.247478 
.248181 
.248883 
.249583 
.250282 
.250980 
.251G77 
.252373 
.2530G7 

9.2537G1 

.254453 

.255144 

.255834 

.256523 

.257211 

.257898 

.258583 

.259268 

.259951 

9.260633 
.261314 
.261994 
.262673 
.263351 
.264027 
.264703 
.265377 
.266051 
.266723 

9.267395 

.268065 

.268734 

.269402 

.270069 

.270735 

.271400 

.272064 

.272726 

.273388 

9.274049 

.274708 

.275367 

.276025 

.276681 

.277337 

.277991 

.278644 

.279297 

.279948 

.280599 

11.93 

11.91 

11.89 

11.87 

11.85 

11.83 
11.81 
11.79 
11.77 
11.75 

11.73 

11.71 

11.69 

11.67 

11.65 

11.63 

11.61 

11.59 

11.58 

11.56 

11.54 

11.52 

11.50 

11.48 

11.46 

11.44 

11.42 

11.41 

11.39 

11.37 

11.35 

11.33 

11.31 

11.30 

11.28 

11.26 

11.24 

11.22 

11.20 

11.19 

11.17 

11.15 

11.13 

11.12 

11.11 

11.08 

11.06 

11.05 

11.03 

11.01 

10.99 

10.98 

10.96 

10.94 

10.92 
10.91 

10.89 

10.87 

10.86 

10.84 

9.993351 

.993329 

.993307 

.993284 

.993262 

.993240 

.993217 

.993195 

.993172 

.993149 

9.993127 

.993104 

.993081 

.993059 

.993036 

.993013 

.992990 

.992967 

.992944 

.992921 

9.992898 

.992875 

.992852 

.992829 

.992806 

.992783 

.992759 

.992736 

.992713 

.992690 

9.992666 

.992643 

.992619 

.992596 

.992572 

.992549 

.992525 

.992501 

.992478 

.992454 

9.992430 

.992406 

.992382 

.992359 

.992335 

.992311 

.992287 

.992263 

.992239 

.992214 

9.992190 

.992166 

.992142 

.992118 

.992093 

.992069 

.992044 

.992020 

.991996 

.991971 

991947 

.37 

.37 

.37 

.37 

.37 

.37 

.38 

.38 

.38 

.38 

.38 

.38 

.38 

.38 

.38 

.38 

.38 

.38 

.38 

.38 

.38 

.38 

.38 

.39 

.39 

.39 

.39 

.39 

.39 

.39 

.39 

.39 

.39 

.39 

.39 

.39 

.39 

.39 

.40 

.40 

.40 

.40 

.40 

.40 

.40 

.40 

.40 

.40 

.40 

.40 

.40 

.40 

.40 

.41 

.41 

.41 

.41 

.41 

.41 

.41 

9.246319 

.247057 

.247794 

.248530 

.249264 

.249998 

.250730 

.251461 

.252191 

.252920 

9.253648 

.254374 

.255100 

.255824 

.256547 

.257269 

.257990 

.258710 

.259429 

.260146 

9.260863 

.261578 

.262292 

.263005 

.263717 

.264428 

.265138 

.265847 

.266555 

.267261 

9.267967 

.268671 

.269375 

.270077 

.270779 

.271479 

.272178 

.272876 

.273573 

.274269 

9.274964 

.275658 

.276351 

.277043 

.277734 

.278424 

.279113 

.279801 

.280488 

.281174 

9.281858 

.282542 

.283225 

.283907 

.284588 

.285268 

.285947 

.286624 

.287301 

.2879/7 

288652 

12.30 
12.28 
12.26 

12.24 
12.22 
12.20 
12.18 
12.17 
12.15 
12.13 

12.11 

12.09 

12.07 

12.05 

12.03 

12.01 

12.00 

11.98 

11.96 

11.94 

11.92 

11.90 

11.89 

11.87 

11.85 

11.83 

11.81 

11.79 

11.78 

11.76 

11.74 

11.72 

11.70 

11.69 

11.67 

11.65 

11.64 

11.62 

11.60 

11.58 

11.57 

11.55 

11.53 

11.51 

11.50 

11.48 

11.47 

11.45 

11.43 

11.41 

11.40 

11.38 

11.36 

11.35 

11.33 

11.31 
11.30 
11.28 
11.26 

11.25 

10.753681 

.752943 

.752206 

.751470 

.750736 

.750002 

.749270 

.748539 

.747809 

.747080 

10.746352 

.745626 

.744900 

.744176 

.743453 

.742731 

.742010 

.741290 

.740571 

.739854 

10.739137 

.738422 

.737708 

.736995 

.736283 

.735572 

.734862 

.734153 

.733445 

.732739 

10.732033 

.731329 

.730625 

.729923 

.729221 

.728521 

.727822 

.727124 

.726427 

.725731 

10.725036 

.724342 

.723649 

.722957 

.722266 

.721576 

.720887 

.720199 

.719512 

.718826 

10.718142 

.717458 

.716775 

.716093 

.715412 

.714732 

.714053 

.713376 

.712699 

.712023 

.711348 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

G 

5 

4 

3 

2 

1 

0 

M. 

Cosine. 

D. 1". 

Sine. 

D. 1 '. 

I Cotang. 

D. 1". 

Tang. 

M. 


100 * 79 ° 















































11 


TABLE IY. LOGARITHMIC BINES, ETC 


51 

168 ° 


M. 

Sine. 

D. 1". 

Cosine. 

D. 1". 

Tang. 

D. 1". 

Cotang. 

M. 

0 

1 

2 

3 

4 

5 

G 

7 

8 
y 

10 

11 

12 

13 

14 

15 
1G 

17 

18 

19 

20 
21 
22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 
3G 

37 

38 

39 

40 

41 

42 

43 

44 

45 
4G 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 

9.280599 

.281248 

.281897 

.282544 

.283190 

.283836 

.284480 

.285124 

.285766 

.286408 

9.287048 

.287688 

.288326 

.288964 

.289600 

.290236 

.290870 

.291504 

.292137 

.292768 

9.293399 

.294029 

.294658 

.295286 

.295913 

.296539 

.297164 

.297788 

.298412 

.299034 

9.299655 

.300276 

.300895 

.301514 

.302132 

.302748 

.303364 

.303979 

.304593 

.305207 

9.305819 

.306430 

.307041 

.307650 

.308259 

.308867 

.309474 

.310080 

.310685 

.311289 

9.311893 

.312495 

.313097 

.313698 

.314297 

.314897 

.315495 

.316092 

.316689 

.317284 

.317879 

10.82 

10.81 

10.79 

10.77 

10.76 

10.74 

10.72 

10.71 

10.69 

10.67 

10.66 

10.64 

10.63 

10.61 

10.59 

10.58 

10.56 

10.55 

10.53 

10.51 

10.50 

10.48 

10.47 

10.45 

10.43 

10.42 

10.40 

10.39 

10.37 

10.36 

10.34 
10.33 
10.31 
10.30 
„ 10.28 
10.26 
10.25 
10.23 
10.22 
10.20 

10.19 

10.17 

10.16 

10.14 

10.13 

10.12 

10.10 

10.09 

10.07 

10.06 

10.04 

10.03 

10.01 

10.00 

9.98 

9.97 

9.96 

9.94 

9.93 

9.91 

9.991947 

.991922 

.991897 

.991873 

.991848 

.991823 

.991799 

.991774 

.991749 

.991724 

9.991699 

.991674 

.991649 

.991624 

.991599 

.991574 

.991549 

.991524 

.991498 

.991473 

9.991448 

.991422 

.991397 

.991372 

.991346 

.991321 

.991295 

.991270 

.991244 

.991218 

9.991193 

.991167 

.991141 

.991115 

.991090 

.991064 

.991038 

.991012 

.990986 

.990960 

9.990934 

.990908 

.990882 

.990855 

.990829 

.990803 

.990777 

.990750 

.990724 

.990697 

9.990671 

.990645 

.990618 

.990591 

.990565 

.990538 

.990511 

.990485 

.990458 

.990431 

.990404 

.41 

.41 

.41 

.41 

.41 

.41 

.41 

.42 

.42 

.42 

.42 

.42 

.42 

.42 

.42 

.42 

.42 

.42 

.42 

.42 

.42 

.42 

.42 

.43 

.43 

.43 

.43 

.43 

.43 

.43 

.43 

.43 

.43 

.43 

.43 

.43 

.43 

.43 

.43 

.43 

.44 

.44 

.44 

.44 

.44 

.44 

.44 

.44 

.44 

.44 

.44 

.44 

.44 

.44 

.44 

.44 

.45 

.45 

.45 

.45 

9.288652 

.289326 

.289999 

.290671 

.291342 

.292013 

.292682 

.293350 

.294017 

.294684 

9.295349 

.296013 

.296677 

.297339 

.298001 

.298662 

.299322 

.299980 

.300638 

.301295 

9.301951 

.302607 

.303261 

.303914 

.304567 

.305218 

.305869 

.306519 

.307168 

.307816 

9.308463 

.309109 

.309754 

.310399 

.311042 

.311685 

.312327 

.312968 

.313608 

.314247 

9.314885 

.315523 

.316159 

.316795 

.317430 

.318064 

.318697 

.319329 

.319961 

.320592 

9.321222 

.321851 

.322479 

.323106 

.323733 

.324358 

.324983 

.325607 

.326231 

.326853 

.327475 

11.23 

11.22 

11.20 

11.18 

11.17 

11.15 

11.14 

11.12 

11.11 

11.09 

11.07 

11.06 

11.04 

11.03 

11.01 

11.00 

10.98 

10.97 

10.95 

10.93 

10.92 

10.90 

10.89 

10.87 

10.86 

10.84 

10.83 

10.81 

10.80 

10.78 

10.77 

10.76 

10.74 

10.73 

10.71 

10.70 

10.68 

10.67 

10.65 

10.64 

10.62 

10.61 

10.60 

10.58 

10.57 

10.55 

10.54 

10.53 

10.51 

10.50 

10.48 

10.47 

10.46 

10.44 

10.43 

10.41 

10.40 

10.39 

10.37 

10.36 

10.711348 

.710674 

.710001 

.709329 

.708658 

.707987 

.707318 

.706650 

.705983 

.705316 

10.704651 

.703987 

.703323 

.702661 

.701999 

.701338 

.700678 

.700020 

.699362 

.698705 

10.698049 

.697393 

.696739 

.696086 

.695433 

.694782 

.694131 

.693481 

.692832 

.692184 

10.691537 

.690891 

.690246 

.689601 

.688958 

.688315 

.687673 

.687032 

.686392 

.685753 

10.685115 

.684477 

.683841 

.683205 

.682570 

.681936 

.681303 

.680671 

.680039 

.679408 

10.678778 

.678149 

.677521 

.676894 

.676267 

.675642 

.675017 

.674393 

.673769 

.673147 

.672525 

60 

59 
58 
57 
56 
55 
54 
53 
52 
51 

60 
49 
48 
47 
46 
45 
44 
43 
42 
41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 

M. 

Cosine. 

D. 1". 

Sine. 

D.l’. 

Cotang. 

D. 1”. 

Tang. 

M. 


101 * 


78 











































TABLE IV. LOGARITHMIC SINES, ETC 


167 


52 


12 ° 


M. 

Sine. 

D.l M . 

Cosine. 

D.l". 

Tang. 

D.l". 

Cotang. 

M. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 
1G 

17 

18 

19 

20 
21 
22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 

9.317879 

.318473 

.319066 

.319658 

.320249 

.320840 

.321430 

.322019 

.322607 

.323194 

9.323780 
.324366 
.324950 
.325534 
.326117 
.326700 
.327281 
.327862 
.328442 
.329021 

9.329599 

.330176 

.330753 

.331329 

.331903 

.332478 

.333051 

.333624 

.334195 

.334767 

9.335337 

.335906 

.336475 

.337043 

.337610 

.338176 

.338742 

.339307 

.339871 

.340434 

9.340996 

.341558 

.342119 

.342679 

.343239 

.343797 

.344355 

.344912 

.345469 

.346024 

9.346579 

.347134 

.347687 

.348240 

.348792 

.349343 

.349893 

.350443 

.350992 

.351540 

.352088 

9.90 

9.88 

9.87 

9.86 

9.84 

9.83 

9.82 

9.80 

9.79 

9.77 

9.76 

9.75 

9.73 

9.72 

9.70 

9.69 

9.68 

9.66 

9.65 

9.64 

9.62 

9.61 

9.60 

9.58 

9.57 

9.56 

9.54 

9.53 

9.52 

9.50 

9.49 

9.48 

9.46 

9.45 

9.44 

9.43 

9.41 

9.40 

9.39 

9.37 

9.36 

9.35 

9.34 

9.32 

9.31 

9.30 

9.29 

9.27 

9.26 

9.25 

9.24 

9.22 

9.21 

9.20 

9.19 

9.17 

9.16 

9.15 

9.14 

9.13 

9.990404 

.990378 

.990351 

.990324 

.990297 

.990270 

.990243 

.990215 

.990188 

.990161 

9.990134 

.990107 

.990079 

.990052 

.990025 

.989997 

.989970 

.989942 

.989915 

.989887 

9.989860 

.989832 

.989804 

.989777 

.989749 

.989721 

.989693 

.989665 

.989637 

.989610 

9.989582 

.989553 

.989525 

.989497 

.989469 

.989441 

.989413 

.989385 

.989356 

.989328 

9.989300 

.989271 

.989243 

.989214 

.989186 

.989157 

.989128 

.989100 

.989071 

.989042 

9.989014 

.988985 

.988956 

.988927 

.988898 

.988869 

.988840 

.988811 

.988782 

.988753 

.988724 

.45 

.45 

.45 

.45 

.45 

.45 

.45 

.45 

.45 

.45 

.45 

.46 

.46 

.46 

.46 

.46 

.46 

.46 

.46 

.46 

.46 

.46 

.46 

.46 

.47 

.47 

.47 

.47 

.47 

.47 

.47 

.47 

.47 

.47 

.47 

.47 

.47 

.47 

.47 

.47 

.47 

.47 

.47 

.47 

.47 

.47 

.48 

.48 

.48 

.48 

.48 

.48 

.48 

.48 

.48 

.48 

.48 

.49 

.49 

.49 

9.327475 

.328095 

.328715 

.329334 

.329953 

.330570 

.331187 

.331803 

.332418 

.333033 

9.333646 

.334259 

.334871 

.335482 

.336093 

.336702 

.337311 

.337919 

.338527 

.339133 

9.339739 

.340344 

.340948 

.341552 

.342155 

.342757 

.343358 

.343958 

.344558 

.345157 

9.345755 

.346353 

.346949 

.347545 

.348141 

.348735 

.349329 

.349922 

.350514 

.351106 

9.351697 

.352287 

.352876 

.353465 

.354053 

.354640 

.355227 

.355813 

.356398 

.356982 

9.357566 

.358149 

.358731 

.359313 

.359893 

.360474 

.361053 

.361632 

.362210 

.362787 

.363364 

10.35 

10.33 

10.32 

10.31 

10.29 

10.28 

10.27 

10.25 

10.24 

10.23 

10.21 

10.20 

10.19 

10.17 

10.16 

10.15 

10.14 

10.12 

10.11 

10.10 

10.08 

10.07 

10.06 

10.05 

10.03 

10.02 

10.01 

10.00 

9.98 

9.97 

9.96 

9.95 

9.93 

9.92 

9.91 

9.90 

9.88 

9.87 

9.86 

9.85 

9.84 

9.82 

9.81 

9.80 

9 79 
9.78 
9.76 
9.75 
9.74 
9.73 

9.72 

9.70 

9.69 

9.68 

9.67 

9.66 

9.65 

9.63 

9.62 

9.61 

10.672525 

.671905 

.671285 

.670666 

.670047 

.669430 

.668813 

.668197 

.667582 

.666967 

10.666354 

.665741 

.665129 

.664518 

.663907 

.663298 

.662689 

.662081 

.661473 

.660867 

10.660261 

.659656 

.659052 

.658448 

.657845 

.657243 

.656642 

.656042 

.655442 

.654843 

10.654245 

.653647 

.653051 

.652455 

.651859 

.651265 

.650671 

.650078 

.649486 

.648894 

10.648303 

.647713 

.647124 

.646535 

.645947 

.645360 

.644773 

.644187 

.643602 

.643018 

10.642434 

.641851 

.641269 

.640687 

.640107 

.639526 

.638947 

.638368 

.637790 

.637213 

.636636 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 

M. 

Cosine. 

D.l". 

Sine. 

D.l'. 

Cotang. 

D.l". 

Tailg. 

M. 


102 ° 











































TABLE IV. LOGARITHMIC SINES, ETC 


53 

166 * 


13 * 



M. 

Siue. 

D.i". 

Cosine. 

D.i •. 

Tang. 

D.I". 

Cotang. 

M. 


0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 
21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 

9.352088 

.352635 

.353181 

.353726 

.354271 

.354815 

.355358 

.355901 

.356443 

.356984 

9.357524 

.358064 

.358603 

.359141 

.359678 

.360215 

.360752 

.361287 

.361822 

.362356 

9.362889 

.363422 

.363954 

.364485 

.365016 

.365546 

.366075 

.366604 

.367131 

.367659 

9.368185 

.368711 

.369236 

.369761 

.370285 

.370808 

.371330 

.371852 

.372373 

.372894 

9.373414 

.373933 

.374452 

.374970 

.375487 

.376003 

.376519 

.377035 

.377549 

.378063 

9.378577 

.379089 

.379601 

.380113 

.380624 

.381134 

.381643 

.382152 

.382661 

.383168 

.383675 

9.11 

9.10 

9.09 

9.08 

9.07 

9.05 

9.04 

9.03 

9.02 

9.01 

8.99 

8.98 

8.97 

8.96 

8.95 

8.94 

8.92 

8.91 

8.90 

8.89 

8.88 

8.87 

8.86 

8.84 

8.83 

8.82 

8.81 

8.80 

8.79 

8.78 

8.76 

8.75 

8.74 

8.73 

8.72 

8.71 

8.70 

8.69 

8.68 

8.66 

8.65 

8.64 

8.63 

8.62 

8.61 

8.60 

8.59 

8.58 

8.57 

8.56 

8.55 

8.53 

8.52 

8.51 

8.50 

8.49 

8.48 

8.47 

8.46 

8.45 

9.988724 

.988695 

.988666 

.988636 

.988607 

.988578 

.988548 

.988519 

.988489 

.988460 

9.988430 

.988401 

.988371 

.988342 

.988312 

.988282 

.988252 

.988223 

.988193 

.988163 

9.988133 

.988103 

.988073 

.988043 

.988013 

.987983 

.987953 

.987922 

.987892 

.987862 

9.987832 

.987801 

.987771 

.987740 

.987710 

.987679 

.987649 

.987618 

.987588 

.987557 

9.987526 

.987496 

.987465 

.987434 

.987403 

.987372 

.987341 

.987310 

.987279 

.987248 

9.987217 

.987186 

.987155 

.987124 

.987092 

.987061 

.987030 

.986998 

.986967 

.986936 

.986904 

.49 

.49 

.49 

.49 

.49 

.49 

.49 

.49 

.49 

.49 

.49 

.49 

.49 

.49 

.50 

.50 

.50 

.50 

.50 

.50 

.50 

.50 

.50 

.50 

.50 

.50 

.50 

.50 

.50 

.50 

.51 

.51 

.51 

.51 

.51 

.51 

.51 

.51 

.51 

.51 

.51 

.51 

.51 

.51 

.52 

.52 

.52 

.52 

.52 

.52 

.52 

.52 

.52 

.52 

.52 

.52 

.52 

.52 

.52 

.52 

9.363364 

.363940 

.364515 

.365090 

.365664 

.366237 

.366810 

.367382 

.367953 

.368524 

9.369094 

.369663 

.370232 

.370799 

.371367 

.371933 

.372499 

.373064 

.373629 

.374193 

9.374756 

.375319 

.375881 

.376442 

.377003 

.377563 

.378122 

.378681 

.379239 

.379797 

9.380354 

.380910 

.381466 

.382020 

.382575 

.383129 

.383682 

.384234 

.384786 

.385337 

9.385888 

.386438 

.386987 

.387536 

.388084 

.388631 

.389178 

.389724 

.390270 

.390815 

9.391360 

.391903 

.392447 

.392989 

.393531 

.394073 

.394614 

.395154 

.395694 

.396233 

.396771 

9.60 

9.59 

9.58 

9.57 

9.55 

9.54 

9.53 

9.52 

9.51 

9.50 

9.49 

9.48 

9.47 

9.45 

9.44 

9.43 

9.42 

9.41 

9.40 

9.39 

9.38 

9.37 

9.36 

9.35 

9.33 

9.32 

9.31 

9.30 

9.29 

9.28 

9.27 

9.26 

9.25 

9.24 

9.23 

9.22 

9.21 

9.20 

9.19 

9.18 

9.17 

9.16 

9.15 

9.14 

9.12 

9.11 

9.10 

9.09 

9.08 

9.07 

9.06 

9.05 

9.04 

9.03 

9.02 

9.01 

9.00 

8.99 

8.98 

8.97 

10.636636 

.636060 

.635485 

.634910 

.634336 

.633763 

.633190 

.632618 

.632047 

.631476 

10.630906 

.630337 

.629768 

.629201 

.628633 

.628067 

.627501 

.626936 

.626371 

.625807 

10.625244 

.624681 

.624119 

.623558 

.622997 

.622437 

.621878 

.621319 

.620761 

.620203 

10.619646 

.619090 

.618534 

.617980 

.617425 

.616871 

.616318 

.615766 

.615214 

.614663 

10.614112 

.613562 

.613013 

.612464 

.611916 

.611369 

.610822 

.610276 

.609730 

.609185 

10.608640 

.608097 

.607553 

.607011 

.606469 

.605927 

.605386 

.604846 

.604306 

.603767 

.603229 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 


M. 

Cosine. 

D.I". 

Sine. 

D.I". 

Cotang. 

D.I". 

Tang. 

M. 




















































54 

14 " 


table iy. logarithmic bijnes, etc. 


105 * 


M. 

Sine. 

D.l". 

Cosine. 

D.l". 

Tang. 

D.l". 

Cotang. 

M. 

0 

1 

2 

3 

4 

5 

C 

7 

8 

9 

10 

u 

12 

13 

14 

15 

16 

17 

18 

19 

20 
21 
22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 
66 

57 

58 

59 

60 

9.383675 

.384182 

.384687 

.385192 

.385697 

.386201 

.386704 

.387207 

.387709 

.388210 

9.388711 

.389211 

.389711 

.390210 

.390708 

.391206 

.391703 

.392199 

.392695 

.393191 

9.393685 

.394179 

.394673 

.395166 

.395658 

.393150 

.396641 

.397132 

.397621 

.398111 

9.398600 

.399088 

.399575 

.400062 

.400549 

.401035 

.401520 

.402005 

.402489 

.402972 

9.403455 

.403938 

.404420 

.404901 

.405382 

.405802 

.406341 

.406820 

.407299 

.407777 

9.408254 

.408731 

.409207 

.409682 

.410157 

.410632 

.411106 

.411579 

.412052 

.412524 

.412996 

8.44 

8.43 

8.42 

8.41 

8.40 

8.39 

8.38 

8.37 

8.36 

8.35 

8.34 

8.33 

8.32 

8.31 

8.30 

8.28 

8.27 

8.26 

8.25 

8.24 

8.23 

8.22 

8.21 

8.20 

8.19 

8.18 

8.17 

8.17 

8.16 

8.15 

8.14 

8.13 

8.12 

8.11 

8.10 

8.09 

8.08 

8.07 

8.06 

8.05 

8.04 

8.03 

8.02 

8.01 

8.00 

7.99 

7.98 

7.97 

7.96 

7.95 

7.94 

7.94 

7.93 

7.92 

7.91 

7.90 

7.89 

7.88 

7.87 

7.86 

9.986904 

.986873 

.986841 

.986809 

.986778 

.986746 

.986714 

.986683 

.986651 

.986619 

9.986587 

.986555 

.986523 

.986491 

.986459 

.986427 

.986395 

.986363 

.986331 

.986299 

9.986266 

.986234 

.986202 

.986169 

.986137 

.986104 

.986072 

.986039 

.986007 

.985974 

9.985942 

.985909 

.985876 

.985843 

.985811 

.985778 

.985745 

.985712 

.985679 

.985646 

9.985613 

.985580 

.985547 

.985514 

.985480 

.985447 

.985414 

.985380 

.985347 

.985314 

9.985280 

.985247 

.985213 

.985180 

.985146 

.985113 

.985079 

.985045 

.985011 

.984978 

.984944 

.53 

.53 

.53 

.53 

.53 

.53 

.53 

.53 

.53 

.53 

.53 

.53 

.53 

.53 

.53 

.53 

.53 

.54 

.54 

.54 

.54 

.54 

.54 

.54 

.54 

.54 

.54 

.54 

.54 

.54 

.54 

.55 

.55 

.55 

.55 

.55 

.55 

.55 

.55 

.55 

.55 

.55 

.55 

.55 

.55 

.55 

.56 

.56 

.56 

.56 

.56 
.56 
.56 
.56 
.56 
.56 
.56 
.56 
.56 
' .5G 

9.396771 

.397309 

.397846 

.398383 

.398919 

.399455 

.399990 

.400524 

.401058 

.401591 

9.402124 

.402656 

.403187 

.403718 

.404249 

.404778 

.405308 

.405836 

.406364 

.40G892 

9.407419 

.407945 

.408471 

.408997 

.409521 

.410045 

.410569 

.411092 

.411615 

.412137 

9.412658 

.413179 

.413699 

.414219 

.414738 

.415257 

.415775 

.416293 

.416810 

.417326 

9.417842 

.418358 

.418873 

.419387 

.419901 

.420415 

.420927 

.421440 

.421952 

.422463 

9.422974 

.423484 

.423993 

.424503 

.425011 

.425519 

.426027 

.426534 

.427041 

.427547 

.428052 

8.96 

8.96 

8.95 

8.94 

8.93 

8.92 

8.91 

8.90 

8.89 

8.88 

8.87 

8.86 

8.85 

8.84 

8.83 

8.82 

8.81 

8.80 

8.79 

8.78 

8.77 

8.76 

8.75 

8.74 

8.74 

8.73 

8.72 

8.71 

8.70 

8.69 

8.68 

8.67 

8.66 

8.65 

8.65 

8.64 

8.63 

8.62 

8.61 

8.60 

8.59 

8.58 

8.57 

8.56 

8.56 

8.55 

8.54 

8.53 

8.52 

8.51 

8.50 

8.49 

8.49 

8.48 

8.47 

8.46 

8.45 

8.44 

8.43 

8.43 

10.603229 

.602691 

.602154 

.601617 

.601081 

.600545 

.600010 

.599476 

.598942 

.598409 

10.597876 

.597344 

.596813 

.596282 

.595751 

.595222 

.594692 

.594164 

.593636 

.593108 

10.592581 

.592055 

.591529 

.591003 

.590479 

.589955 

.589431 

.588908 

.588385 

.587863 

10.587342 

.586821 

.586301 

.585781 

.585262 

.584743 

.584225 

.583707 

.583190 

.582674 

10.582158 

.581642 

.581127 

,580613 

.580099 

.579585 

.579073 

.578560 

.578048 

.577537 

10.577026 

.576516 

.576007 

.575497 

.574989 

.574481 

.573973 

.573466 

.572959 

.572453 

.571948 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 

M. 

Cosine, 

D.l". 

Sine. 

D.l". 

Cotang. 

D.l". 

Tang. 

M. 


104 ° 75 » 







































15 


TABLE IV. LOGAllITHMIC SINES, ETC 


65 


104 ° 


M. 

Sine. 

D.l'". 

Cosine. 

D.l". 

Tang. 

D.l . 

Cotang. 

M. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 
21 
22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 

9.412996 
.413467 
.413938 
.414408 
.414878 
.415347 
.415815 
.416283 
.416751 
.417217 

9.417684 

.418150 

.418615 

.419079 

.419544 

.420007 

.420470 

.420933 

.421395 

.421857 

9.422318 

.422778 

.423238 

.423697 

.424156 

.424615 

.425073 

.425530 

.425987 

.426443 

9.426899 

.427354 

.427809 

.428263 

.428717 

.429170 

.429623 

.430075 

.430527 

.430978 

9.431429 

.431879 

.432329 

.432778 

.433226 

.433675 

.434122 

.434569 

.435016 

.435462 

9.435908 

.436353 

.436798 

.437242 

.437686 

.438129 

.438572 

.439014 

.439456 

.439897 

.440338 

7.85 

7.84 

7.84 

7.83 

7.82 

7.81 

7.80 

7.79 

7.78 

7.77 

7.76 

7.75 

7.75 

7.74 

7.73 

7.72 

7.71 

7.70 

7.69 

7.68 

7.67 

7.67 

7.66 

7.65 

7.64 

7.63 

7.62 

7.61 

7.61 

7.60 

7.59 

7.58 

7.57 

7.56 

7.55 

7.54 

7.53 

7.52 

7.52 

7.51 

7.50 

7.49 

7.49 

7.48 

7.47 

7.46 

7.45 

7.44 

7.44 

7.43 

7.42 

7.41 

7.40 

7.40 

7.39 

7.38 

7.37 

7.36 

7.36 

7.35 

9.984944 

.984910 

.984876 

.984842 

.984808 

.984774 

.984740 

.984706 

.984672 

.984637 

9.984603 

.984569 

.984535 

.984500 

.984466 

.984432 

.984397 

.984363 

.984328 

.984294 

9.984259 

.984224 

.984190 

.984155 

.984120 

.984085 

.984050 

.984015 

.983981 

.983946 

9.983911 

.983875 

.983840 

.983805 

.983770 

.983735 

.983700 

.983664 

.983629 

.983594 

9.983558 

.983523 

.983487 

.983452 

.983416 

.983381 

.983345 

.983309 

.983273 

.983238 

9.983202 

.983166 

.983130 

.983094 

.983058 

.983022 

.982986 

.982950 

.982914 

.982878 

.982842 

.57 

.57 

.57 

.57 

.57 

.57 

.57 

.57 

.57 

.57 

.57 

.57 

.57 

.57 

.57 

.58 

.58 

.58 

.58 

.58 

.58 

.58 

.58 

.58 

.58 

.58 

.58 

.58 

.58 

.58 

.58 

.58 

.59 

.59 

.59 

.59 

.59 

.59 

.59 

.59 

.59 

.59 

.59 

.59 

.59 

.59 

.59 

.59 

.60 

.60 

.60 

.60 

.60 

.60 

.60 

.60 

.60 

.60 

.60 

.60 

9.428052 

.428558 

.429062 

.429566 

.430070 

.430573 

.431075 

.431577 

.432079 

.432580 

9.433080 

.433580 

.434080 

.434579 

.435078 

.435576 

.436073 

.436570 

.437067 

.437563 

9-438059 

.438554 

.439048 

.439543 

.440036 

.440529 

.441022 

.441514 

.442006 

.442497 

9.442988 

.443479 

.443968 

.444458 

.444947 

.445435 

.445923 

.446411 

.446898 

.447384 

9.447870 

.448356 

.448841 

•449326 

.449810 

.450294 

.450777 

.451260 

.451743 

.452225 

9.452706 

.453187 

.453668 

.454148 

.454628 

.455107 

.455586 

.456064 

.456542 

.457019 

.457496 

8.42 

8.41 

8.40 

8.39 

8.38 

8.38 

8.37 

8.36 

8.35 

8.34 

8.33 

8.33 

8.32 

8.31 

8.30 

8.29 

8.28 

8.28 

8.27 

8.26 

8.25 

8.24 

8.24 

8.23 

8.22 

8.21 

8.20 

8.20 

8.19 

8.18 

8.17 

8.16 

8.16 

8.15 

8.14 

8.13 

8.13 

8.12 

8.11 

8.10 

8.09 

8.09 

8.08 

8.07 

8.06 

8.06 

8.05 

8.04 

8.03 

8.03 

8.02 

8.01 

8.00 

8.00 

7.99 

7.98 

7.97 

7.97 

7.96 

7.95 

10.571948 

.571442 

.570938 

.570434 

.569930 

.569427 

.568925 

.568423 

.567921 

.567420 

10.566920 

.566420 

.565920 

.565421 

.564922 

.564424 

.563927 

.563430 

.562933 

.562437 

10.561941 

.561446 

.560952 

.560457 

.559964 

.559471 

.558978 

.558486 

.557994 

.557503 

10.557012 

.556521 

.556032 

.555542 

.555053 

.554565 

.554077 

.553589 

.553102 

.552616 

10.552130 

.551644 

.551159 

.550674 

.550190 

.549706 

.549223 

.548740 

.548257 

.547775 

10.547294 

.546813 

.546332 

.545852 

.545372 

.544893 

.544414 

.543936 

.543458 

.542981 

.542504 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 

M. 

Cosine. 

D.l '. 

Sine. 

D.l’’. 

Cotang. 

D.l". 

Tang. 

M. 




































TABLE IY. LOGARITHMIC SINES, ETC, 


163 


56 


16 ° 


M. 

Sine. 

D.l". 

Cosine. 

D.l" 

Tang. 

D.l". 

Cotang. 

M. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 
10 

17 

18 

19 

20 
21 
22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 
30 

37 

38 

39 

40 

41 

42 

43 
41 
45 
40 

47 

48 

49 

50 

51 

52 

53 

54 

55 
50 

57 

58 

59 
00 

9.440338 

.440778 

.441218 

.441058 

.442090 

.442535 

.442973 

.443410 

.443847 

.444284 

9.444720 

.445155 

.445590 

.446025 

.440459 

.446893 

.447326 

.447759 

.418191 

.448623 

9.449054 

.449485 

.449915 

.450345 

.450775 

.451204 

.451632 

.452000 

.452488 

.452915 

9.453342 

.453708 

.454194 

.454619 

.455044 

.455469 

.455893 

.456310 

.456739 

.457162 

9.457584 

.458006 

.458427 

.458848 

.459268 

.459688 

.400108 

.460527 

.460946 

.461364 

9.461782 

.462199 

.462616 

.463032 

.463448 

.463864 

.464279 

.464694 

.465108 

.465522 

.465935 

7.34 

7.33 

7.32 

7.31 

7.31 

7.30 

7.29 

7.28 

7.27 

7.27 

7.26 

7.25 

7.24 

7.24 

7.23 

7.22 

7.21 

7.20 

7.20 

7.19 

7.18 

7.17 

7.17 

7.16 

7.15 

7.14 

7.13 

7.13 

7.12 

7.11 

7.10 

7.10 

7.09 

7.08 

7.07 

7.07 

7.06 

7.05 

7.04 

7.04 

7.03 

7.02 

7.01 

7.01 

7.00 

6.99 

6.98 

6.98 

6.97 

6.96 

6.96 

6.95 

6.94 

6.93 

6.93 

6.92 

6.91 

6.90 

6.90 

6.89 

9.982842 

.982805 

.982769 

.982733 

.982696 

.982660 

.982624 

.982587 

.982551 

.982514 

9.982477 

.982441 

.982404 

.982367 

.982331 

.982294 

.982257 

.982220 

.982183 

.982146 

9.982109 

.982072 

.982035 

.981998 

.981961 

.981924 

.981886 

.981849 

.981812 

.981774 

9.981737 

.981700 

.981662 

.981625 

.981587 

.981549 

.981512 

.981474 

.981436 

.981399 

9.981361 
.981323 
.981285 
.981247 
.981209 
.981171 
.981133 
.981095 
.981057 . 
.981019 

9.980981 

.980942 

.980904 

.980866 

.980827 

.980789 

.980750 

.980712 

.980673 

.980635 

.980596 

.60 

.60 

.61 

.61 

.61 

.61 

.61 

.61 

.61 

.61 

.61 

.61 

.61 

.61 

.61 

.61 

.61 

.62 

.62 

.62 

.62 

.62 

.62 

.62 

.62 

.62 

.62 

.62 

.62 

.62 

.62 

.62 

.63 

.63 

.63 

.63 

.63 

.63 

.63 

.63 

.63 

.63 

.63 

.63 

.63 

.63 

.63 

.64 

.64 

.64 

.64 

.64 

.64 

.64 

.64 

.64 

.64 

.64 

.64 

.64 

9.457496 

.457973 

.458449 

.458925 

.459400 

.459875 

.460349 

.460823 

.461297 

.461770 

9.462242 

.462714 

.463186 

.463658 

.464129 

.464599 

.465069 

.465539 

.466008 

.466476 

9.466945 
.467413 
.467880 
.468347 
.468814 
.469280 
.469746 
.470211 
.470676 
.471141 

9.471605 
.472068 
.472532 
.472995 
.473457 
.473919 
.474381 
.474842 
.475303 
.475763 

9.476223 

.476683 

.477142 

.477601 

.478059 

.478517 

.478975 

.479432 

.479889 

.480345 

9.480801 

.481257 

.481712 

.482167 

.482621 

.483075 

.483529 

.483982 

.484435 

.484887 

.485339 

7.94 

7.94 

7.93 

7.92 

7.91 

7.91 

7.90 

7.89 

7.88 

7.88 

7.87 

7.86 

7.86 

7.85 

7.84 

7.83 

7.83 

7.82 

7.81 

7.81 

7.80 

7.79 

7.78 

7.78 

7.77 

7.76 

7.76 

7.75 

7.74 

7.74 

7.73 

7.72 

7.71 

7.71 

7.70 

7.69 

7.69 

7.68 

7.67 

7.67 

7.66 

7.65 

7.65 

7.64 

7.63 

7.63 

7.62 

7.61 

7.61 

7.60 

7.59 

7.59 

7.58 

7.57 

7.57 

7.56 

7.55 

7.55 

7.54 

7.53 

10.542504 

.542027 

.511551 

.541075 

.540600 

.540125 

.539651 

.539177 

.538703 

.538230 

10.537758 

.537286 

.536814 

.536342 

.535871 

.535401 

.534931 

.534461 

.533992 

.533524 

10.533055 

.532587 

.532120 

.531653 

.531186 

.530720 

.530254 

.529789 

.529324 

.528859 

10.528395 

.527932 

.527468 

.527005 

.526543 

.526081 

.525619 

.525158 

.524697 

.524237 

10.523777 

.523317 

.522858 

.522399 

.521941 

.521483 

.521025 

.520568 

.520111 

.519655 

10.519199 

.518743 

.518288 

.517833 

.517379 

.516925 

.516471 

.516018 

.515565 

.515113 

.514661 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

32 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 

M. 

Cosine. 

D.l". 

Sine. 

D.l". 

Cotang. 

D.l". 

Tang. 

M. 


106 ° 


73 

































IT 


TABLE IV. LOGARITHMIC SINES, ETC 


67 


162 ° 


M. 

Sine. 

D.l". 

Cosine. 

D.l”. 

Tang. 

D.l”. 

Cotang. 

M. 

0 

1 

2 

3 

4 

5 

G 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 
21 
22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 

9.465935 

.466348 

.466761 

.467173 

.467585 

.467996 

.468407 

.468817 

.469227 

.469637 

9,470046 

.470455 

.470863 

.471271 

.471679 

.472086 

.472492 

.472898 

.473304 

.473710 

9.474115 

.474519 

.474923 

.475327 

.475730 

.476133 

.476536 

.476938 

.477340 

.477741 

9.478142 

.478542 

,.478942 
.479342 
.479741 
.480140 
.480539 
.480937 
.481334 
.481731 

9.482128 

.482525 

.482921 

.483316 

.483712 

.484107 

.484501 

.484895 

.485289 

.485682 

9.486075 

.486467 

.486860 

,487251 

.487643 

.488034 

.488424 

.488814 

.489204 

.489593 

.489982 

6.88 

6.88 

6.87 

6.86 

6.85 

6.85 

6.84 

6.83 

6.83 

6.82 

6.81 

6.81 

6.80 

6.79 

6.78 

6.78 

6.77- 

6.76 

6.76 

6.75 

6.74 

6.74 

6.73 

6 72 
6.72 
6.71 
6.70 
6.69 
6.69 
6.68 

6.67 

6.67 

6.66 

6.65 

6.65 

6.64 

6.63 

6.63 

6.62 

6.61 

6.61 

6.60 

6.59 

6.59 

6.58 

6.57 

6.57 

6.56 

6.55 

6.55 

6.54 

6.54 

6.53 

6.52 

6.52 

6.51 

6.50 

6.50 

6.49 

6.48 

9.980596 

.980558 

.980519 

.980480 

.980442 

.980403 

.980364 

.980325 

.980286 

.980247 

9.980208 

.980169 

.980130 

.980091 

.980052 

.980012 

.979973 

.979934 

.979895 

.979855 

9.979816 

.979776 

.979737 

.979697 

.979658 

.979618 

.979579 

.979539 

.979499 

.979459 

9.979420 

.979380 

.979340 

.979300 

.979260 

.979220 

.979180 

.979140 

.979100 

.979059 

9.979019 

.978979 

.978939 

.978898 

.978858 

.978817 

.978777 

.978736 

-.978696 

.978655 

9.978615 

.978574 

.978533 

.978493 

.978452 

.978411 

.978370 

.978329 

.978288 

.978247 

.978206 

.64 

.64 

.65 

.65 

.65 

.65 

.65 

.65 

.65 

.65 

.65 

.65 

.65 

.65 

.65 

.65 

.65 

.66 

.66 

.66 

.66 

.66 

.66 

.66 

.66 

.66 

.66 

.66 

.66 

.66 

.66 

.66 

.67 

.67 

.67 

.67 

.67 

.67 

.67 

.67 

.67 

.67 

.67 

.67 

.67 

.67 

.67 

.68 

.68 

.68 

.68 

.68 

.68 

.68 

.68 

.68 

.68 

.68 

.68 

.68 

9.485339 

.485791 

.486242 

.486693 

.487143 

.487593 

.488043 

.488492 

.488941 

.489390 

9.489838 

.490286 

.490733 

.491180 

.491627 

.492073 

.492519 

.492965 

.493410 

.493854 

9.494299 

.494743 

.495186 

.495630 

.496073 

.496515 

.496957 

.497399 

.497841 

.498282 

9.498722 

.499163 

.499603 

.500042 

.500481 

.500920 

.501359 

.501797 

.502235 

.502672 

9.503109 

.503546 

.503982 

.504418 

.504854 

.505289 

.505724 

.506159 

.506593 

.507027 

9.507460 

.507893 

.508326 

.508759 

.509191 

.509622 

.510054 

.510485 

.510916 

.511346 

.511776 

7.53 

7.52 

7.51 

7.51 

7.50 

7.50 

7.49 

7.48 

7.48 

7.47 

7.46 

7.46 

7.45 

7.44 

7.44 

7.43 

7.43 

7.42 

7.41 

7.41 

7.40 

7.39 

7.39 

7.38 

7.38 

7.37 

7.36 

7.36 

7.35 

7.34 

7.34 

7.33 

7.33 

7.32 

7.31 

7.31 

7.30 

7.30 

7.29 

7.28 

7.28 

7.27 

7.27 

7.26 

7.25 

7.25 

7.24 

7.24 

7.23 

7.23 

7.22 

7.21 

7.21 

7.20 

7.20 

7.19 

7.18 

7.18 

7.17 

7.17 

10.514661 

.514209 

.513758 

.513307 

.512857 

.512407 

.511957 

.511508 

.511059 

.510610 

10.510162 

.509714 

.509267 

.508820 

.508373 

.507927 

.507481 

.507035 

.506590 

•506146 

10.505701 

.505257 

.504814 

.504370 

.503927 

.503485 

.503043 

.502601 

.502159 

.501718 

10.501278 

.500837 

.500397 

.499958 

.499519 

.499080 

.498641 

.498203 

.497765 

.497328 

10.496891 

.496454 

.496018 

.495582 

.495146 

.494711 

.494276 

.493841 

.493407 

.492973 

10.492540 

.492107 

.491674 

.491241 

.490809 

.490378 

.489946 

.489515 

.489084 

.488654 

.488224 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 

M. 

Cosine. 

D.l”. 

Sine. 

D.l”. 

Cotang. 

D.l”. 

Tang. 

M. 


107 * E , 72- 































68 TABLE IV. LOGARITHMIC SINES, ETC, 


161 


18° 


M. 

Sine. 

D.l . 

Cosine. 

D.l". 

Tang. 

D.l". 

Cotang, 

M. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 
1G 

17 

18 

19 

20 
21 

22 

23 

24 

25 
20 

27 

28 

29 

30 

31 

32 

33 

34 

35 
30 

37 

38 

39 

40 

41 

42 

43 

44 

45 
40 

47 

48 

49 

50 

51 

52 

53 

54 

55 
50 

57 

58 

59 
00 

9.489982 

.490371 

.490759 

.491147 

.491535 

.491922 

.492308 

.492095 

.493081 

.493400 

9.493851 

.494230 

.494021 

.495005 

.495388 

.495772 

.490154 

.490537 

.490919 

.497301 

9.497082 

.498004 

.498444 

.498825 

.499204 

.499584 

.499903 

.500342 

.500721 

.501099 

9.501470 

.501854 

.502231 

.502007 

.502984 

.503300 

.503735 

.504110 

.501485 

.504800 

9.505234 

.505008 

.505981 

.506354 

.506727 

.507099 

.507471 

.507843 

.508214 

.508585 

9.508956 

.509320 

.509696 

.510065 

.510434 

.510803 

.511172 

.511540 

.511907 

.512275 

.512642 

6.48 

6.47 

6.40 

6.40 
0.45 
6.45 
6.44 
6.43 
6.43 
6.42 

6.41 
6.41 
6.40 
6.39 
6.39 
0.38 
6.38 
6.37 
6.36 
6.36 

6.35 

6.34 

6.34 

6.33 

6.33 

0.32 

6 31 
6.31 
6.30 
6.30 

6.29 

6.28 

6.28 

6.27 

6.27 

6.26 

6.25 

6.25 

6.24 

6.24 

6.23 

0.22 

6.22 

6.21 

6.21 

(T.20 

6.19 

6.19 

6.18 

6.18 

6.17 

6.16 

6.16 

6.15 

6.15 

6.14 

6.14 

6.13 

6.12 

6.12 

9.978206 

.978165 

.978124 

.978083 

.978042 

.978001 

.977959 

.977918 

.977877 

.977835 

9.977794 

.977752 

.977711 

.977669 

.977628 

.977586 

.977544 

.977503 

.977461 

.977419 

9.977377 

.977335 

.977293 

.977251 

.977209 

.977167 

.977125 

.977083 

.977041 

.976999 

9.976957 

.976914 

.976872 

.976830 

.976787 

.976745 

.976702 

.976660 

.976617 

.976574 

9.976532 

.976489 

.970446 

.976404 

.976361 

.976318 

.976275 

.976232 

.976189 

.976146 

9.976103 

.976060 

.976017 

.975974 

.975930 

.975887 

.975844 

.975800 

.975757 

.975714 

.975670 

.68 

.69 

.69 

.69 

.69 

.69 

.69 

.69 

.69 

.69 

.69 

.69 

.69 

.69 

.69 

.69 

.70 

.70 

.70 

.70 

.70 

.70 

.70 

.70 

.70 

.70 

.70 

.70 

.70 

.70 

.70 

.71 

.71 

.71 

.71 

.71 

.71 

.71 

.71 

.71 

.71 

.71 

.71 

.71 

.71 

.72 

.72 

.72 

.72 

.72 

.72 

.72 

.72 

.72 

.72 

.72 

.72 

.72 

.72 

.72 

9.511776 

.512206 

.512635 

.513064 

.513493 

.513921 

.514349 

.514777 

.515204 

.515631 

9.516057 

.516484 

.516910 

.517335 

.517761 

.518185 

.518610 

.519034 

.519458 

.519882 

9.520305 

.520728 

.521151 

.521573 

.521995 

.522417 

.522838 

.523259 

.523680 

.524100 

9.524520 

.524939 

.525359 

.525778 

.526197 

.526615 

.527033 

.527451 

.527868 

.528285 

9.528702 

.529119 

.529535 

.529950 

.530366 

.530781 

.531196 

.531611 

.532025 

.532439 

9.532853 

.533266 

.533679 

.534092 

.534504 

.534916 

.535328 

.535739 

.536150 

.536561 

.536972 

7.16 

7.16 

7.15 

7.14 

7.14 

7.13 

7.13 

7.12 

7.12 

7.11 

7.10 

7.10 

7.09 

7.09 

7.08 

7.08 

7.07 

7.07 

7.06 

7.05 

7.05 

7.04 

7.04 

7.03 

7.03 

7.02 

7.02 

7.01 

7.01 

7.00 

6.99 

6.99 

6.98 

6.98 

6.97 

6.97 

6.96 

6.96 

6.95 

6.95 

6.94 

6.94 

6.93 

6.93 

6.92 

6.91 

6.91 

6.90 

6.90 

6.89 

6.89 

6.88 

6.88 

6.87 

6.87 

6.86 

6.86 

6.85 

6.85 

6.84 

10.488224 

.487794 

.487365 

.486936 

.486507 

.486079 

.485651 

.485223 

.484796 

.484369 

10.483943 

.483516 

.483090 

.482665 

.482239 

.481815 

.481390 

.480966 

.480542 

.480118 

10.479695 

.479272 

.478849 

.478427 

.478005 

.477583 

.477162 

.476741 

.476320 

.475900 

10.475480 

.475061 

.474641 

.474222 

.473803 

.473385 

.472967 

.472549 

.472132 

.471715 

10.471298 

.470881 

.470465 

.470050 

.4696:14 

.469219 

.468804 

.468389 

.467975 

.467561 

10.467147 

.466734 

.466321 

.465908 

.465496 

.46.5084 

.464672 

.464261 

.463850 

.463439 

.463028 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 : 

i 

0 

M. 

Cosine. 

D.l". 

Sine. 

D.l". 

I Cotang. 

D.l". 

Tang. 

M. 


108 ° 


71 ' 
















































19 


TABLE IY. LOGARITHMIC SINES, ETC, 


59 


160 ° 


M. 

Sine. 

D. 1". 

Cosine. 

D.l". 

Tang. 

D. 1". 

Cotang. 

M. 

0 

1 

2 

3 

4 

5 
G 

7 

8 
9 

10 

11 

12 

13 

14 

15 
1G 

17 

18 

19 

20 
21 
22 

23 

24 

25 
2G 

27 

28 

29 

30 

31 

32 

33 

34 

35 
3G 

37 

38 

39 

40 

41 

42 

43 

44 

45 
4G 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 
GO 

9.512G42 
.513009 
.513375 
.513741 
.514107 
.514472 
.514837 
.515202 
.5155G6 
.515930 

9.516294 

.516657 

.517020 

.517382 

.517745 

.518107 

.518468 

.518829 

.519190 

.519551 

9.519911 

.520271 

.520631 

.520990 

.521349 

.521707 

.522066 

.522424 

.522781 

.523138 

9.523495 
.523852 
.524208 
.524564 
.524?20 
.525275 
.525630 
.525984 
.526339 
.526693 

9.527046 

.527400 

.527753 

.528105 

.528458 

.528810 

.529161 

.529513 

.529864 

.530215 

9.530565 

.530915 

.531265 

.531614 

.531963 

.532312 

.532661 

.533009 

.533357 

.533704 

.534052 

6.11 

6.11 

6.10 

6.09 

6.09 

6.08 

6.08 

6.07 

6.07 

6.06 

6.05 

6.05 

6.04 

6.04 

6.03 

6.03 

6.02 

6.02 

6.01 

6.00 

6.00 

5.99 

5.99 

5.98 

5.98 

5.97 

5.97 

5.96 

5.95 

5.95 

5.94 

5.94 

5.93 

5.93 

5.92 

5.92 

5.91 

5.90 

5.90 

5.89 

5.89 

5.88 

5.88 

5.87 

5.87 

5.86 

5.86 

5.85 

5.85 

5.84 

5.83 

5.83 

5.82 

5.82 

5.81 

5.81 

5.80 

5.80 

5.79 

5.79 

9.975670 

.975627 

.975583 

.975539 

.975496 

.975452 

.975408 

.975365 

.975321 

.975277 

9.975233 

.975189 

.975145 

.975101 

.975057 

.975013 

.974969 

.974925 

.974880 

.974836 

9.974792 

.974748 

.974703 

.974659 

.974614 

.974570 

.974525 

.974481 

.974436 

.974391 

9.974347 

.974302 

.974257 

.974212 

.974167 

.974122 

.974077 

.974032 

.973987 

.973942 

9.973897 

.973852 

.973807 

.973761 

.973716 

.973671 

.973625 

.973580 

.973535 

.973489 

9.973444 

.973398 

.973352 

.973307 

.973261 

.973215 

.973169 

.973124 

’ .973078 
.973032 
.972986 

.73 

.73 

.73 

.73 

.73 

.73 

.73 

.73 

.73 

.73 

.73 

.73 

.73 

.73 

.73 

.74 

.74 

.74 

.74 

.74 

.74 

.74 

.74 

.74 

.74 

.74 

.74 

.74 

.74 

.75 

.75 

.75 

.75 

..75 

.75 

.75 

.75 

.75 

.75 

.75 

.75 

.75 

.75 

.75 

.76 

.76 

.76 

.76 

.76 

.76 

.76 

.76 

.76 

.76 

.76 

.76 

.76 

776 

.77 

.77 

9.536972 

.537382 

.537791 

.538202 

.538611 

.539020 

.539429 

.539837 

.540245 

.540653 

9.541061 

.541468 

.541875 

.542281 

.542688 

.543094 

.543499 

.543905 

.544310 

.544715 

9.545119 

.545524 

.545928 

.546331 

.546735 

.547138 

.547540 

.547943 

.548345 

.548747 

9.549149 

.549550 

.549951 

.550352 

.550752 

.551152 

.551552 

.551952 

.552351 

.552750 

9.553149 

.553548 

.553946 

.554344 

.554741 

.555139 

.555536 

.555933 

.556329 

.556725 

9.557121 

.557517 

.557913 

.558308 

.558702 

.559097 

.559491 

.559885 

.560279 

.560673 

.561066 

6.84 

6.83 

6.83 

6.82 

6.82 

6.81 

6.81 

6.80 

6.80 

6.79 

6.79 

6.78 

6.78 

6.77 

6.77 

6.76 

6.76 

6.75 

6.75 

6.74 

6.74 

6.73 

6.73 

6.72 

6.72 

6.71 

6.71 

6.70 

6.70 

6.69 

6.69 

6.68 

6.68 

6.67 

6.67 

6.67 

6.66 

6.66 

6.65 

6.65 

6.64 

6.64 

6.63 

6.63 

6.62 

6.62 

6.61 

6.61 

6.60 

6.60 

6.59 

6.59 

6.59 

6.58 

6.58 

6.57 

6.57 

6.56 

6.56 

6.55 

10.463028 

.462618 

.462209 

.461798 

.461389 

.460980 

.460571 

.460163 

.459755 

.459347 

10.458939 

.458532 

.458125 

.457719 

.457312 

.456906 

.456501 

.456095 

.455690 

.455285 

10.454881 

.454476 

.454072 

.453669 

.453265 

.452862 

.452460 

.452057 

.451655 

.451253 

10.450851 

.450450 

.450049 

.449648 

.449248 

.448848 

.448448 

.448048 

.447649 

.447250 

10.446851 

.446452 

.446054 

.445656 

.445259 

.444861 

.444464 

.444067 

.443671 

.443275 

10.442879 

.442483 

.442087 

.441692 

.441298 

.440903 

.440509 

.440115 

.439721 

.439327 

.438934 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 

M. 

Cosine. 

D. 1 

Sine. 

D.l". 

Cotang. 

D. 1". 

Tang. 

M. 


109 ' 70 ° 









































TABLE IV. LOGAEITHMIC SINES, ETC, 


159 


CO 

20 ° 


M. 

Sine. 

D.l”. 

Cosine. 

D.l". 

0 

1 

2 

3 

4 

5 

6 

7 

8 
.9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 
21 
22 

23 

24 

25 
2G 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 
68 

59 

60 

9.534052 

.534399 

.534745 

.535092 

.535438 

.535783 

.536129 

.536474 

.536818 

.537163 

9.537507 

.537851 

.538194 

.538538 

.538880 

.539223 

.539565 

.539907 

.540249 

.540590 

9.540931 

.541272 

.541613 

.541953 

.542293 

.542632 

.542971 

.543310 

.543649 

.543987 

9.544325 

.544663 

.545000 

.545338 

.545674 

.546011 

.546347 

.546683 

.547019 

.547354 

9.547689 

.548024 

.548359 

.548693 

.549027 

.549360 

.549693 

.550026 

.550359 

.550692 

9.551024 

.551356 

.551687 

.552018 

.552349 

.552680 

.553010 

.553341 

.553670 

.554000 

.554329 

5.78 

5.78 

5.77 

5.77 

5.76 

5.76 

5.75 

5.75 

6.74 

5.74 

5.73 

5.73 

5.72 

5.71 

5.71 

5.70 

5.70 

5.69 

5.69 

5.68 

5.68 

5.67 

6.67 
5.66 
5.66 
5.65 
5.65 
5.64 
5.64 
5.63 

5.63 
5.62 
5.62 
5.61 
5.61 
5. GO 
5.60 

5.59 

6.59 

6.58 

5.58 

6.57 

5.57 
6.56 
6.56 
5.55 
5.55 
5.55 
5.54 
5.54 

6.53 
• 6.53 

5.52 
5.52 

6.51 

5.51 
5.50 
5.50 

5.49 

6.49 

9.972986 

.972940 

.972894 

.972848 

.972802 

.972755 

.972709 

.972663 

.972617 

.972570 

9.972524 

.972478 

.972431 

.972385 

.972338 

.972291 

.972245 

.972198 

.972151 

.972105 

9.972058 

.972011 

.971964 

.971917 

.971870 

.971823 

.971776 

.971729 

.971682 

.971635 

9.971588 

.971540 

.971493 

.971446 

.971398 

.971351 

.971303 

.971256 

.971208 

.971161 

9.971113 

.971066 

.971018 

.970970 

.970922 

.970874 

.970827 

.970779 

.970731 

.970683 

9.970635 

.970586 

.970538 

.970490 

.970442 

.970394 

.970345 

.970297 

.970249 

.970200 

.970152 

.77 

.77 

.77 

.77 

.77 

.77 

.77 

.77 

.77 

.77 

.77 

.77 

.78 

.78 

.78 

.78 

.78 

.78 

.78 

.78 

.78 

.78 

.78 

.78 

.78 

.78 

.78 

.79 

.79 

.79 

.79 

.79 

.79 

.79 

.79 

.79 

.79 

.79 

.79 

.79 

.79 

.80 

.80 

.80 

.80 

.80 

.80 

.80 

.80 

.80 

.80 

.80 

.80 

.80 

.80 

.81 

.81 

.81 

.81 

.81 

M. 

Cosine. 

D.l". 

Sine. 

D.l". 


110 ° 


Tang. 

D.l". 

C-otang. 

M. 

9.561066 

6.55 

6.54 

6.54 

6.54 

6.53 

6.53 

6.52 

6.52 

6.51 

6.51 

10.438934 

60 

.561459 

.438541 

59 

.561851 

.438149 

58 

.562244 

.437756 

57 

.562636 

.437364 

56 

.563028 

.436972 

55 

.563419 

.436581 

54 

.563811 

.436189 

53 

.564202 

.435798 

52 

.564592 

.435408 

51 

9.564983 

6.50 

6.50 

6.50 

6.49 

6.49 

6.48 

6.48 

6.47 

6.47 

6.46 

10.435017 

50 

.565373 

.434627 

49 

.565763 

.434237 

48 

.566153 

.433847 

47 

.566542 

.433458 

46 

.566932 

.433068 

45 

.567320 

.432680 

44 

.567709 

.432291 

43 

.568098 

.431902 

42 

.568486 

.431514 

41 

9.568873 

6.46 

6.46 

6.45 

6.45 

6.44 

6.44 

6.43 

6.43 

6.43 

6.42 

10.431127 

40 

.569261 

.430739 

39 

.569648 

.430352 

38 

.570035 

.429965 

37 

.570422 

.429578 

36 

.570809 

.429191 

35 

.571195 

.428805 

34 

.571581 

.428419 

33 

.571967 

.428033 

32 

.572352 

.427648 

31 

9.572738 

6.42 

6.41 

6.41 

6.40 

6.40 

6.40 

6.39 

6.39 

6.38 

6.38 

10.427262 

30 

.573123 

.573507 

.426877 

.426493 

29 

28 

.573892 

.426108 

27 

.574276 

.425724 

26 

.574660 

.425340 

25 

.575044 

.424956 

24 

.575427 

.424573 

23 

.575810 

.424190 

22 

.576193 

.423807 

21 

9.576576 

6.37 

6.37 

6.37 

6.36 

6.36 

6.35 

6.35 

6.34 

6.34 

6.34 

10.423424 

20 

.576958 

.577341 

.423041 

.422659 

19 

18 

.577723 

.422277 

17 

.578104 

.421896 

16 

.578486 

.421514 

15 

.578867 

.421133 

14 

.579248 

.579629 

.420752 

.420371 

13 

12 

.580009 

.419991 

11 

9.580389 

6.33 

6.33 

6.32 

6.32 

6.32 

6.31 

6.31 

6.30 

6.30 

6.30 

10.419611 

10 

.580769 

.419231 

9 

.581149 

.418851 

8 

.581528 

.418472 

7 

.581907 

.418093 

6 

.582286 

.417714 

5 

.582665 

.417335 

4 

.583043 

.416957 

3 

.583422 

.416578 

2 

.583800 

.416200 

1 

.584177 

.415823 

0 

Cotang. 

D.l". 

Tang. 

M. 


69 0 




































TABLE IY. LOGARITHMIC SINES, ETC 


61 


21 * 158 ° 


M. 

Sine. 

D.l . 

Cosine. 

D.l 

Tang. 

D.l". 

Cotang. 

M. 

0 

9.554329 

5.48 

9.970152 

.81 

9.584177 

6.29 

10.415823 

60 

1 

.554658 

5.48 

.970103 

.81 

.584555 

6.29 

.415445 

59 

2 

.554987 

5.47 

.970055 

.81 

.584932 

6.28 

.415068 

58 

3 

.555315 

5.47 

.970006 

.81 

.585309 

6.28 

.414691 

57 

4 

.555643 

6.46 

.969957 

.81 

.585686 

6.28 

.414314 

56 

5 

.555971 

5.46 

.969909 

.81 

.586062 

6.27 

.413938 

55 

G 

.556299 

5.45 

.969860 

.81 

.586439 

6.27 

.413561 

54 

7 

.556626 

5.45 

.969811 

.81 

.586815 

6.26 

.413185 

53 

8 

.556953 

5.44 

.969762 

.81 

.587190 

6.26 

.412810 

52 

9 

.557280 

5.44 

.969714 

.81 

.587566 

6.26 

.412434 

51 

10 

9.557606 

5.44 

9.969665 

.82 

9.587941 

6.25 

10.412059 

50 

11 

.557932 

5.43 

•969616 

.82 

.588316 

6.25 

.411684 

49 

12 

.558258 

5.43 

.969567 

.82 

.588691 

6.24 

.411309 

48 

13 

.558583 

5.42 

.969518 

.82 

.589066 

6.24 

.410934 

47 

14 

.558909 

6.42 

.969469 

.82 

.589440 

6.24 

.410560 

46 

15 

.559234 

6.41 

.969420 

.82 

.589814 

6.23 

.410186 

45 

16 

.559558 

5.41 

.969370 

.82 

.590188 

6.23 

.409812 

44 

17 

.559883 

5.40 

.969321 

.82 

.590562 

6.22 

.409438 

43 

18 

.560207 

5.40 

.969272 

.82 

.590935 

6.22 

.409065 

42 

19 

.560531 

5.39 

.969223 

.82 

.591308 

6.22 

.408692 

41 

20 

9.560855 

5.39 

9.969173 

.82 

9.591681 

6.21 

10.408319 

40 

21 

.561178 

5.38 

.969124 

.82 

.592054 

6.21 

.407946 

39 

22 

.561501 

5.38 

.969075 

.82 

.592426 

6.20 

.407574 

38 

23 

.561824 

5.37 

.969025 

.82 

.592798 

6.20 

.407202 

37 

24 

.562146 

5.37 

.968976 

.83 

.593171 

6.20 

.406829 

36 

25 

.562468 

5.37 

.968926 

.83 

.593542 

6.19 

.406458 

35 

26 

.562790 

5.36 

.968877 

.83 

.593914 

6.19 

.406086 

34 

27 

.563112 

5.36 

.968827 

.83 

.594285 

6.18 

.40^715 

33 

28 

.563433 

5.35 

.968777 

.83 

.594656 

6.18 

.405344 

32 

29 

.563755 

5.35 

.968728 

.83 

.595027 

6.18 

.404973 

31 

30 

9.564075 

5.34 

9.968678 

.83 

9.595398 

6.17 

10.404602 

30 

31 

.564396 

5.34 

.968628 

.83 

.595768 

6.17 

.404232 

29 

32 

.564716 

5.33 

.968578 

.83 

.596138 

6.16 

.403862 

28 

33 

.565036 

5.33 

.968528 

.83 

.596508 

6.16 

.403492 

27 

34 

.565356 

5.32 

.968479 

.83 

.596878 

6.16 

.403122 

26 

35 

.565676 

5.32 

.968429 

.83 

.597247 

6.15 

.402753 

25 

36 

.565995 

5.32 

.968379 

.83 

.597616 

6.15 

.402384 

24 

37 

.566314 

5.31 

.968329 

.83 

.597985 

6.15 

.402015 

23 

38 

.566632 

5.31 

.968278 

.84 

.598354 

6.14 

.401646 

22 

39 

.566951 

5.30 

.968228 

.84 

.598722 

6.14 

.401278 

21 

40 

9.567269 

5.30 

9.968178 

.84 

9.599091 

6.13 

10.400909 

20 

41 

.567587 

5.29 

.968128 

.84 

.599459 

6 13 

.400541 

19 

42 

.567904 

5.29 

.968078 

.84 

.599827 

6.13 

.400173 

18 

43 

.568222 

5.28 

.968027 

.84 

.600194 

6.12 

.399806 

17 

44 

.568539 

5.28 

.967977 

.84 

.600562 

6.12 

.399438 

16 

45 

.568856 

5.28 

.967927 

.84 

.600929 

6.12 

.399071 

15 

46 

.569172 

5.27 

.967876 

.84 

.601296 

6.11 

.398704 

14 

47 

.569488 

5.27 

.967826 

.84 

.601662 

6.11 

.398338 

13 

48 

.569804 

5.26 

.967775 

.84 

.602029 

6.10 

.397971 

12 

49 

.570120 

5.26 

.967725 

.84 

.602395 

6.10 

.397605 

11 

50 

9.570435 

5.25 

9 967674 

.84 

9.602761 

6.10 

10.397239 

10 

51 

.570751 

5.25 

.967624 

.84 

.603127 

6.09 

.396873 

9 

52 

.571066 

5.24 

.967573 

.85 

.603493 

6.09 

.396507 

8 

53 

.571380 

5.24 

.967522 

.85 

.603858 

6.09 

.396142 

7 

54 

.571695 

5.24 

.967471 

.85 

.604223 

6.08 

.395777 

6 

55 

.572009 

5.23 

.967421 

.85 

.604588 

6.08 

.395412 

5 

56 

.572323 

5.23 

.967370 

.85 

.604953 

6.07 

.395047 

4 

57 

.572636 

5.22 

.967319 

.85 

.605317 

6.07 

.394683 

3 

58 

.572950 

5.22 

.967268 

.85 

.605682 

6.07 

.394318 

2 

59 

.573263 

5.21 

.967217 

.85 

.606046 

6.06 

.393954 

1 

1,60 

.573575 


.967166 


.606410 


.393590 

0 

M. 

Cosine. 

D.l". 

Sine. 

D.l". 

Cotang. 

D.l ".i_Tang. 

M. 


Ill* 68* 







































TABLE IV. LOGARITHMIC SINES, ETC, 


62 


22 " 157 ° 


M. 

Sine. 

D.l". 

Cosine. 

D.l". 

Tang. 

D.l". 

Cotang. 

M. 

0 

1 

2 

3 

4 

5 

r. 

7 

8 

3 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 
21 
22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 
3-1 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 
CO 

9.573575 

.573888 

.574200 

.574512 

.574824 

.575136 

.575447 

.575758 

.576069 

.576379 

9.576689 

.576999 

.577309 

.577618 

.577927 

.578236 

.578545 

.578853 

.579162 

.579470 

9.579777 

.580085 

.580392 

.580699 

.581005 

.581312 

.581618 

.581924 

.582229 

.582535 

9.582840 

.583145 

.583449 

.583754 

.584058 

.584361 

.584665 

.584968 

.585272 

.585574 

9.585877 

.586179 

.586482 

.586783 

.587085 

.587386 

.587688 

.587989 

.588289 

.588590 

9.588890 

.589190 

.589489 

.589789 

.590088 

.590387 

.590686 

.590984 

.591282 

.591580 

.591878 

5.21 

5.20 

5.20 

5.20 

5.19 

5.19 

6.18 

5.18 

5.17 

5.17 

5.17 

5.16 

5.16 

5.15 

5.15 

5.14 

5.14 

5.14 

5.13 

5.13 

5.12 

5.12 

5.11 

5.11 

5.11 

5.10 

5.10 

5.09 

5.09 

5.09 

5.08 

5.08 

5.07 

5.07 

5.06 

5.06 

5.06 

5.05 

5.05 

5.04 

5.04 

5.04 

6.03 

5.03 

5.02 

5.02 

6.01 

5.01 

5.01 

5.00 

5.00 

4.99 

4.99 

4.99 

4.98 

4.98 

4.97 

4.97 

4.97 

4.96 

9.967166 

.967115 

.967064 

.967013 

.966961 

.966910 

.966859 

.966808 

.966756 

.966705 

9.966653 

.966602 

.966550 

.966499 

.966447 

.966395 

.966344 

.966292 

.966240 

.966188 

9.966136 

.966085 

.966033 

.965981 

.965928 

.965876 

.965824 

.965772 

.965720 

.965668 

9.965615 

.965563 

.965511 

.965458 

.965406 

.965353 

.965301 

.965248 

.965195 

.965143 

9.965090 

.965037 

.964984 

.964931 

.964879 

.964826 

.964773 

.964720 

.964666 

.964613 

9.964560 

.964507 

.964454 

.964400 

.964347 

.964294 

.964240 

.964187 

.964133 

.964080 

.964026 

.85 

.85 

.85 

.85 

.85 

.85 

.86 

.86 

.86 

.86 

.86 

.86 

.86 

.86 

.86 

.86 

.86 

.86 

.86 

.86 

.87 

.87 

.87 

.87 

.87 

.87 

.87 

.87 

.87 

.87 

.87 

.87 

.87 

.87 

.88 

.88 

.88 

.88 

.88 

.88 

.88 

.88 

.88 

.88 

.88 

.88 

.88 

.88 

.89 

.89 

.89 

.89 

.89 

.89 

.89 

.89 

.89 

.89 

.89 

.89 

9.606410 

.606773 

.607137 

.607500 

.607863 

.608225 

.608588 

.608950 

.609312 

.609674 

9.610036 

.610397 

.610759 

.611120 

.611480 

.611841 

.612201 

.612561 

.612921 

.613281 

9.613641 

.614000 

.614359 

.614718 

.615077 

.615435 

.615793 

.616151 

.616509 

.616867 

9.617224 

.617582 

.617939 

.618295 

.618652 

.619008 

.619364 

.619721 

.620076 

.620432 

9.620787 

.621142 

.621497 

.621852 

.622207 

.622561 

.622915 

.623269 

.623623 

.623976 

9.624330 

.624683 

.625036 

.625388 

.625741 

.626093 

.626445 

.626797 

.627149 

.627501 

.627852 

6.06 

6.06 

6.05 

6.05 

6.05 

6.04 

6.04 

6.03 

6.03 

6.03 

6.02 

6.02 

6.02 

6.01 

6.01 

6.01 

6.00 

6.00 

6.00 

5.99 

5.99 

5.98 

5.98 

5.98 

5.97 

5.97 

5.97 

5.96 

5.96 

5.96 

5.95 

5.95 

5.95 

5.94 

5.94 

5.94 

5.93 

5.93 

5.93 

5.92 

5.92 

5.92 

5.91 

5.91 

5.91 

5.90 

5.90 

5.90 

5.89 

5.89 

5.89 

5.88 

5.88 

5.88 

5.87 

5.87 

5.87 

5.86 

5.86 

5.86 

10.393590 

.393227 

.392863 

.392500 

.392137 

.391775 

.391412 

.391050 

.390688 

.390326 

10.389964 

.389603 

.389241 

.388880 

.388520 

.388159 

.387799 

.387439 

.387079 

.386719 

10.386359 

.386000 

.385641 

.385282 

.384923 

.384565 

.384207 

.383849 

.383491 

.383133 

10.382776 

.382418 

.382061 

.381705 

.381348 

.380992 

.380636 

.380280 

.379924 

.379568 

10.379213 

.378858 

.378503 

.378148 

.377793 

.377439 

.377085 

.376731 

.376377 

.376024 

10.375670 

.375317 

.374964 

.374612 

.374259 

.373907 

.373555 

.373203 

.372851 

.372499 

.372148 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 

M. 

Cosine. 

D.l'. 

Sine. 

D.l". 

Cot ail g. 

D.l". 

Tang. 

M. 











































23 


TABLE IV. LOGARITHMIC SINES, ETC, 


63 


ir>6 0 


M. 

Sine. 

D.l". 

Cosine. 

D.l" 

Tang. 

D.l’. 

Cotang. 

M. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 
21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 
GO 

9.591878 

.592176 

.592473 

.592770 

.593067 

.593363 

.593659 

.593955 

.594251 

.594547 

9.594842 
.595137 
.595432 
.595727 
.596021 
.596315 
.596609 
.596903 
.597196 
.597490 

9.597783 

.598075 

.598368 

.598660 

.598952 

.599244 

.599536 

.599827 

.600118 

.600409 

9.600700 

.600990 

.601280 

.601570 

.601860 

.602150 

.602439 

.602728 

.603017 

.603305 

9.603594 

.603882 

.604170 

.604457 

.604745 

.605032 

.605319 

.605606 

.605892 

.606179 

9.606465 

.606751 

.607036 

.607322 

.607607 

.607892 

.608177 

.608461 

.608745 

.609029 

.609313 

4.96 

4.95 

4.95 

4.95 

4.94 

4.94 

4.93 

4.93 

4.93 

4.92 

4.92 

4.91 

4.91 

4.91 

4.90 

4.90 

4.89 

4.89 

4.89 

4.88 

4.88 

4.88 

4.87 

4.87 

4.86 

4.86 

4.86 

4.85 

4.85 

4.84 

4.84 

4.84 

4.83 

4.83 

4.83 

4.82 

4.82 

4.81 

4.81 

4.81 

4.80 

4.80 

4.79 

4.79 

4.79 

4.78 

4.78 

4.78 

4.77 

4.77 

4.76 

4.76 

4.76 

4.75 

4.75 

4.74 

4.74 

4.74 

4.73 

4.73 

9.964026 

.963972 

.963919 

.963865 

.963811 

.963757 

.963704 

.963650 

.963596 

.963542 

9.963488 

.963434 

.963379 

.963325 

.963271 

.963217 

.963163 

.963108 

.963054 

.962999 

9.962945 

.962890 

.962836 

.962781 

.962727 

.962672 

.962617 

.962562 

.962508 

.962453 

9.962398 

.962343 

.962288 

.962233 

.962178 

.962123 

.962067 

.962012 

.961957 

.961902 

9.961846 

.961791 

.961735 

.961680 

.961624 

.961569 

.961513 

.961458 

.961402 

.961346 

9.961290 

.961235 

.961179 

.961123 

.961067 

.961011 

.960955 

.960899 

.960843 

.960786 

.960730 

.89 

.89 

.90 

.90 

.90 

.90 

.90 

.90 

.90 

.90 

.90 

.90 

.90 

.90 

.90 

.90 

.91 

.91 

.91 

.91 

.91 

.91 

.91 

.91 

.91 

.91 

.91 

.91 

.91 

.92 

.92 

.92 

.92 

.92 

.92 

.92 

.92 

.92 

.92 

.92 

.92 

.92 

.92 

.93 

.93 

.93 

.93 

.93 

.93 

.93 

.93 

.93 

.93 

.93 

.93 

.93 

.93 

.94 

.94 

.94 

9.627852 

.628203 

.628554 

.628905 

.629255 

.629606 

.629956 

.630306 

.630656 

.631005 

9.631355 

.631704 

.632053 

.632401 

.632750 

.633098 

.633447 

.633795 

.634143 

.634490 

9.634838 

.635185 

.635532 

.635879 

.636226 

.636572 

.636919 

.637265 

.637611 

.637956 

9.638302 

.638647 

.638992 

.639337 

.639682 

.640027 

.640371 

.640716 

.641060 

.641404 

9.641747 

.642091 

.642434 

.642777 

.643120 

.643463 

.643806 

.644148 

.644490 

.644832 

9.645174 

.645516 

.645857 

.646199 

.646540 

.646881 

.647222 

.647562 

.647903 

.648243 

.648583 

5.85 

5.85 

5.85 

5.84 

5.84 

5.84 

5.83 

5.83 

5.83 

5.82 

5.82 

5.82 

5.81 

5.81 

5.81 

5.80 

5.80 

5.80 

5.79 

5.79 

5.79 

5.78 

5.78 

5.78 

5.78 

5.77 

5.77 

5.77 

5.76 

5.76 

6.76 
5.75 
5.75 
5.75 
5.74 
5.74 
5.74 
5.73 
5.73 
5.73 

5.73 

5.72 

5.72 

5.72 

5.71 

5.71 

5.71 

5.70 

5.70 

6.70 

5.69 

5.69 

5.69 

5.69 

5.68 

5.68 

5.68 

5.67 

5.67 

5.67 

10.372148 

.371797 

.371446 

.371095 

.370745 

.370394 

.370044 

.369694 

.369344 

.368995 

10.368645 

.368296 

.367947 

.367599 

.367250 

.366902 

.366553 

.366205 

.365857 

.365510 

10.365162 

.364815 

.364468 

.364121 

.363774 

.363428 

.363081 

.362735 

.362389 

.362044 

10.361698 

.361353 

.361008 

.360663 

.360318 

.359973 

.359629 

.359284 

.358940 

.358596 

10.358253 

.357909 

.357566 

.357223 

.356880 

.356537 

.356194 

.355852 

.355510 

.355168 

10.354826 

.354484 

.354143 

.353801 

.353460 

.353119 

.352778 

.352438 

.352097 

.351757 

.351417 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 

M. 

Cosine. 

D.l 

Sine. 

D.l". 

Cotang. 

D.l". 

Tang. 

M. 


113 


66 

































TABLE IV. LOGARITHMIC SINES, ETC. 


155 


64 


24 * 


M. 

Sine. 

D.l”. 

Cosine. 

D.l”. 

Tang. 

D.l”. 

Cotang. 

M. 


0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 
21 
22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 
69 
60 

9.609313 

.609597 

.609880 

.610164 

.610447 

.610729 

.611012 

.611294 

.611576 

.611858 

9.612140 

.612421 

.612702 

.612983 

.613264 

.613545 

.613825 

.614105 

.614385 

.614665 

9.614944 

.615223 

.615502 

.615781 

.616060 

.616338 

.616616 

.616894 

.617172 

.617450 

9.617727 

.618004 

.618281 

.618558 

.618834 

.619110 

.619386 

.619662 

.619938 

.620213 

9.620488 

.620763 

.621038 

.621313 

.621587 

.621861 

.622135 

.622409 

.622682 

.622956 

9.623229 

.623502 

.623774 

.624047 

.624319 

.624591 

.624863 

.625135 

.625406 

.625677 

.625948 

4.73 

4.72 

4.72 

4.72 

4.71 

4.71 

4.71 

4.70 

4.70 

4.69 

4.69 

4.69 

4.68 

4.68 

4.68 

4.67 

4.67 

4.67 

4.66 

4.66 

4.65 

4.65 

4.65 

4.64 

4.64 

4.64 

4.63 

4.63 

4.63 

4.62 

4.62 

4.61 

4.61 

4.61 

4.60 

4.60 

4.60 

4.59 

4.59 

4.59 

4.58 

4.58 

4.58 

4.57 

4.57 

4.57 

4.56 

4.56 

4.56 

4.55 

4.55 

4.54 

4.54 

4.54 

4.53 

4.53 

4.53 

4.52 

4.52 

4.52 

9.960730 

.960674 

.960618 

.960561 

.960505 

.960448 

.960392 

.960335 

.960279 

.960222 

9.960165 

.960109 

.960052 

.959995 

.959938 

.959882 

.959825 

.959768 

.959711 

.959654 

9.959596 

.959539 

.959482 

.959425 

.959368 

.959310 

.959253 

.959195 

.959138 

.959081 

9.959023 

.958965 

.958908 

.958850 

.958792 

.958734 

.958677 

.958619 

.958561 

.958503 

9.958445 

.958387 

.958329 

.958271 

.958213 

.958154 

.958096 

.958038 

.957979 

.957921 

9.957863 

.957804 

.957746 

.957687 

.957628 

.957570 

.957511 

.957452 

.957393 

.957335 

.957276 

.94 

.94 

.94 

.94 

.94 

.94 

.94 

.94 

.94 

.94 

.95 

.95 

.95 

.95 

.95 

.95 

.95 

.95 

.95 

.95 

.95 

.95 

.95 

.95 

.96 

.96 

.96 

.96 

.96 

.96 

.96 

.96 

.96 

.96 

.96 

.96 

.96 

.97 

.97 

.97 

.97 

.97 

.97 

.97 

.97 

.97 

.97 

.97 

.97 

.97 

.97 

.98 

.98 

.98 

.98 

.98 

.98 

.98 

.98 

.98 

9.048583 

.648923 

.649263 

.649602 

.649942 

.650281 

.650620 

.650959 

.651297 

.651636 

9.651974 

.652312 

.652650 

.652988 

.653326 

.653663 

.654000 

.654337 

.654674 

.655011 

9.055348 

.655084 

.656020 

.656356 

.656692 

.657028 

.657364 

.657699 

.658034 

.658369 

9.658704 

.659039 

.659373 

.659708 

.600042 

.660376 

.660710 

.661043 

.661377 

.661710 

9.662043 

.062376 

.662709 

.663042 

.663375 

.663707 

.664039 

.664371 

.664703 

.665035 

9.065366 

.665697 

.666029 

.666360 

.666691 

.667021 

.667352 

.667682 

.668013 

.668343 

.668672 

5.67 

5.66 

5.66 

5.66 

5.65 

5.65 

5.65 

5.64 

5.64 

5.64 

5.64 

5.63 

5.03 

5.63 

5.62 

5.62 

5.62 

5.62 

5.61 

5.61 

5.61 

5.61 

5.60 

5.60 

6.60 
5.59 
5.59 
5.59 
5.58 
5.58 

5.58 

5.58 

5.57 

5.57 

5.57 

6.56 

5.56 
5.56 
5.56 
5.55 

5.55 

5.55 

5.54 

5.54 

5.54 

5.54 

5.53 

5.53 

5.53 

5.53 

5.52 

5.52 

5.52 

5.51 

5.51 

5.51 

5.51 

5.50 

5.50 

5.50 

10.351417 

.351077 

.350737 

.350398 

.350058 

.349719 

.349380 

.349041 

.348703 

.348364 

10.348026 
.347688 
.347350 
.347012 
.346674 
.346337 
.340000 
.345663 
.345326 
.344989 

10.344652 

.344316 

.343980 

.343644 

.343308 

.342972 

.342036 

.342301 

.341966 

.341631 

10.341296 

.340961 

.340627 

.340292 

.339958 

.339624 

.339290 

.338957 

.338023 

.338290 

10.337957 

.337624 

.337291 

.336958 

.336625 

.336293 

.335961 

.335629 

.335297 

.334965 

10.334634 

.334303 

.333971 

.333640 

.333309 

.332979 

.332648 

.332318 

.331987 

.331657 

.331328 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11* 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 


M. 

Cosine. 

D.l". 

Sine. 

D.l”. 

Cotang. 

D.l”. 

Tang. 

M. 



114 ® 


65 ® 












































25 


TABLE IV. LOGARITHMIC SINES, ETC. 


65 


154 ’ 


M. 

Sine. 

D. 1". 

Cosine. 

D. 1”. 

Tang. 

D. 1”. 

Cotang. 

M. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 
1G 

17 

18 

19 

20 
21 
22 

23 

24 

25 
2G 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 
4G 

47 

48 

49 

50 

51 

52 

53 

54 

55 
5G 

57 

58 
53 
CO 

9.625948 

.626219 

.626490 

.626760 

.627030 

.627300 

.627570 

.627840 

.628109 

.628378 

9.628647 

.628916 

.629185 

.629453 

.629721 

.629989 

.630257 

.630524 

.630792 

.631059 

9.631326 

.631593 

.631859 

.632125 

.632392 

.632658 

.632923 

.633189 

.633454 

.633719 

9.633984 

.634249 

.634514 

.634778 

.635042 

.635306 

.635570 

.635834 

.636097 

.636360 

9.636623 

.636886 

.637148 

.637411 

.637673 

.637935 

.638197 

.638458 

.638720 

.638981 

9.639242 

.639503 

.639764 

.640024 

.640284 

.640544 

.640804 

.641064 

.641324 

.641583 

.641842 

4.51 

4.51 

4.51 

4.50 

4.50 

4.50 

4.49 

4.49 

4.49 

4.48 

4.48 

4.48 

4.47 

4.47 

4.47 

4.46 

4.46 

4.46 

4.45 

4.45 

4.45 

4.44 

4.44 

4.44 

4.43 

4.43 

4.43 

4.42 

4.42 

4.42 

4.41 

4.41 

4.41 

4.40 

4.40 

4.40 

4.39 

4.39 

4.39 

4.38 

4.38 

4.38 

4.37 

4.37 

4.37 

4.36 

4.36 

4.36 

4.35 

4.35 

4.35 

4.34 

4.34 

4.34 

4.33 

4.33 

4.33 

4.32 

4.32 

4.32 

9.957276 

.957217 

.957158 

.957099 

.957040 

.956981 

.956921 

.956862 

.956803 

.956744 

9.956684 

.956625 

.956566 

.956506 

.956447 

.956387 

.956327 

.956268 

.956208 

.956148 

9.956089 

.956029 

.955969 

.955909 

.955849 

.955789 

.955729 

.955669 

955609 

.955548 

9.955488 

.955428 

.955368 

.955307 

.955247 

.955186 

.955126 

.955065 

.955005 

.954944 

9.954883 

.954823 

.954762 

.954701 

.954640 

.954579 

.954518 

.954457 

.954396 

.954335 

9.954274 

.954213 

.954152 

.954090 

.954029 

.953968 

.953906 

.953845 

.953783 

.953722 

.953660 

.98 

.98 

.98 

.98 

.99 

.99 

.99 

.99 

.99 

.99 

.99 

.99 

.99 

.99 

.99 

.99 

.99 

.99 

1.00 

1.00 

1.00 

1.00 

1.00 

1.00 

1.00 

1.00 

1.00 

1.00 

1.00 

1.00 

1.00 

1.01 

1.01 

1.01 

1.01 

1.01 

1.01 

1.01 

1.01 

1.01 

1.01 

1.01 

1.01 

1.01 

1.02 

1.02 

1.02 

1.02 

1.02 

1.02 

1.02 

1.02 

1.02 

1.02 

1.02 

1.02 

1.02 

1.03 

1.03 

1.03 

9.668673 

.669002 

.669332 

.669661 

.669991 

.670320 

.670649 

.670977 

.671306 

.671634 

9.671963 

.672291 

.672619 

.672947 

.673274 

.673602 

.673929 

.674257 

.674584 

.674910 

9.675237 

.675564 

.675890 

.676216 

.676543 

.676869 

.677194 

.677520 

.677846 

.678171 

9.678496 

.678821 

.679146 

.679471 

.679795 

.680120 

.680444 

.680768 

.681092 

.681416 

9.681740 

.682063 

.682387 

.682710 

.683033 

.683356 

.683679 

.684001 

.684324 

.684646 

9.6849C8 

.685290 

.685612 

.685934 

.686255 

.686577 

.686898 

.687219 

.687540 

.687861 

.688182 

5.50 

5.49 

5.49 

5.49 

5.49 

5.48 

5.48 

5.48 

5.47 

5.47 

5.47 

5.47 

5.46 

5.46 

5.46 

5.46 

5.45 

5.45 

5.45 

5.45 

6.44 

5.44 
5.44 
5.44 
5.43 
5.43 
5.43 
5.42 
5.42 
5.42 

5.42 

5 41 
5.41 
5.41 
5.41 
5.40 
5.40 
5.40 
5.40 
5.39 

5.39 

5.39 

5.39 

5.38 

5.38 

5.38 

5.38 

5.37 

5.37 

5.37 

5.37 

5.36 

5.36 

5.36 

5.36 

5.35 

5.35 

5.35 

5.35 

5.35 

10.331327 

.330998 

.330668 

.330339 

.330009 

.329680 

.329351 

.329023 

.328694 

.328366 

10.328037 

.327709 

.327381 

.327053 

.326726 

.326398 

.326071 

325743 

.325416 

.325090 

10.324763 

.324436 

.324110 

.323784 

.323457 

.323131 

.322806 

.322480 

.322154 

.321829 

10.321504 

.321179 

.320854 

.320529 

.320205 

.319880 

.319556 

.319232 

.318908 

.318584 

10.318260 

.317937 

.317613 

.317290 

.316967 

.316644 

.316321 

.315999 

.315676 

.315354 

10.315032 

.314710 

.314388 

.314066 

.313745 

.313423 

.313102 

.312781 

.312460 

.312139 

.311818 

60 

59 

58 

67 

56 

55 

64 

53 

62 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 

M. 

Cosine. 

D. 1'. 

Sine. 

D. 1". 

Cotang. 

D. 1". 

Tang. 

M. 


115 * 64 * 
































66 TABLE IV. LOGARITHMIC SINES, ETC. 


153 ° 


M. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 
10 

17 

18 

19 

20 
21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 

Sine. 

D.l”., 

Cosine. 

D.l". 

Tang. 

D.l". 

Cotang. 

M. 

9.641842 

.642101 

.642360 

.642618 

.642877 

.643135 

.643393 

.643650 

.643908 

.644165 

9.644423 

.644680 

.644936 

.645193 

.645450 

.645706 

.645962 

.646218 

.646474 

.646729 

9.646984 

.647240 

.647494 

.647749 

.648004 

.648258 

.648512 

.648766 

.649020 

.649274 

9.649527 

.649781 

.650034 

.650287 

.650539 

.650792 

.651044 

.651297 

.651549 

.651800 

9.652052 

.652304 

.652555 

.652806 

.653057 

.653308 

.653558 

.653808 

.654059 

.654309 

9.654558 

.654808 

.655058 

.655307 

.655556 

.655805 

.656054 

.656302 

.656551 

.656799 

.657047 

4.32 

4.31 

4.31 

4.31 

4.30 

4.30 

4.30 

4.29 

4.29 

4.29 

4.28 

4.28 

4.28 

4.27 

4.27 

4.27 

4.26 

4.26 

4.26 

4.26 

4.25 

4.25 

4.25 

4 24 
4.24 
4.24 
4.23 
4.23 
4.23 
4.22 

4.22 

4.22 

4.22 

4.21 

4.21 

4.21 

4.20 

4.20 

4.20 

4.19 

4 19 
4.19 
4.18 
4.18 
4.18 
4.18 
4.17 
4.17 
4.17 
4.16 

4.16 

4.16 

4.15 

4.15 

4.15 

4.15 

4.14 

4.14 

4.14 

4.13 

9.953660 
.953599 
.953537 
.933475 
.953413 
.953352 
.953^90 
.953228 
.953166 
.953104 

9.953042 

.952980 

.952918 

.952855 

.952793 

.952731 

.952669 

.952606 

.952544 

.952481 

9.952419 

.952356 

.952294 

.952231 

.952168 

.952106 

.952043 

.951980 

.951917 

.951854 

9.951791 

.951728 

.951665 

.951602 

.951539 

.951476 

.951412 

.951349 

.951286 

.951222 

9.951159 

.951096 

.951032 

.950968 

.950905 

.950841 

.950778 

.950714 

.950650 

.950586 

9.950522 
.950458 
.950394 
.950330 
.950266 
.950202 
.950138 
.950074 
.950010 
. 9491)45 
.949881 

1.03 

1.03 

1.03 

1.03 

1.03 

1.03 

1.03 

1.03 

1 03 
1.03 

1.03 

1.04 

1.04 

1.04 

1.04 

1.04 

1.04 

1.04 

1.04 

1.04 

1.04 

1 04 
1.04 
1.04 
1.05 
1.05 
1.05 
1.05 
1.05 
1.05 

1.05 

1.05 

1.05 

1.05 

1.05 

1.05 

1.05 

1 06 
1.06 
1.06 

1.06 

1.06 

1.06 

1.06 

1.06 

1.06 

1.06 

1.06 

1.06 
1.06 

1.07 

1.07 

1.07 

1.07 

1.07 

1.07 

1.07 

1.07 

1.07 

1.07 

9.688182 

.688502 

.688823 

.689143 

.689463 

.689783 

.690103 

.690423 

.690742 

.691062 

9.691381 

.691700 

.692019 

.692338 

.692656 

.692975 

.693293 

.693612 

.693930 

.694248 

9.694566 

.694883 

.695201 

.695518 

.695836 

.696153 

.696470 

.696787 

.697103 

.697420 

9.697736 

.698053 

.698369 

.698685 

.699001 

.699316 

.699632 

.699947 

.700263 

.700578 

9.700893 

.701208 

.701523 

.701837 

.702152 

.702466 

.702781 

.703095 

.703409 

.703723 

9.704036 
.704350 
.704663 
.704977 
.705290 
.705603 
.705916 
.706228 
. 70(5541 
.706854 
.707166 

5.34 

5.34 

5.34 

5.34 

5.33 

5.33 

5.33 

5.33 

5.32 

5.32 

5.32 

5.32 

5.31 

5.31 

5.31 

5.31 

5.30 

5.30 

5.30 

5.30 

5.29 

5.29 

5.29 

5.29 

5.29 

5.28 

5.28 

5.28 

5.28 

5.27 

5.27 

5.27 

5.27 

5.26 

5.26 

5.26 

5.26 

5.26 

5.25 

5.25 

5.25 

5.25 

5.24 

5.24 

5.24 

5.24 

5.24 

5.23 

5.23 

5.23 

5.23 

5.22 

5.22 

5.22 

5.22 

5.22 

5.21 

5.21 

5.21 

5.21 

10.311818 

.311498 

.311177 

.310857 

.310537 

.310217 

.309897 

.309577 

.309258 

.308938 

10.308619 

.308300 

.307981 

.307662 

.307344 

.307025 

.306707 

.306388 

.306070 

.305752 

10.305434 

.305117 

.304799 

.304482 

.304164 

.303847 

.303530 

.303213 

.302897 

.302580 

10.302264 

.301947 

.301631 

.301315 

.300999 

.300684 

.300368 

.300053 

.299737 

299422 

10.299107 

.298792 

.298477 

.298163 

.297848 

.297534 

.297219 

.296905 

.296591 

.296273 

10.295964 

.295650 

.295337 

.295023 

.294710 

.294397 

.204084 

.293772 

.293459 

.293146 

.292834 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 

M. 

Cosine. 

D.l". 

Sine. 

D.l". 

Cotang. 

D.l". 

Tang. 

M. 


116 ° 63 e 

































27 


TABLE IV. LOGARITHMIC SINES, ETC, 


67 


152 ' 


M. 

Sine. 

D.l". 

Cosine. 

D.l". 

Tang.. 

D.l". 

Cotang. 

M. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 
21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 

9.657047 

.657295 

.657542 

.657790 

.658037 

.658284 

.658531 

.658778 

.659025 

.659271 

9.659517 

.659763 

.660009 

.660255 

.660501 

.660746 

.660991 

.661236 

.661481 

.661726 

9.661970 

.662214 

.662459 

.662703 

.662946 

.663190 

.663433 

.663677 

.663920 

.664163 

9.664406 

.664648 

.664891 

.665133 

.665375 

.665617 

.665859 

.666100 

.666342 

.666583 

9.666824 

.667065 

.667305 

.667546 

.667786 

.668027 

.668267 

.668506 

.668746 

.668986 

9.669225 
.669464 
.669703 
.669942 
.670181 
.670419 
.670658 
.670896 
.671134 
.671372 
.671609 

4.13 

4.13 

4.12 

4.12 

4.12 

4.12 

4.11 

4.11 

4.11 

4.10 

4.10 

4.10 

4.10 

4.09 

4.09 

4.09 

4.08 

4.08 

4.08 

4.08 

4.07 

4.07 

4.07 

4.06 

4.06 

4.06 

4.05 

4.05 

4.05 

4.05 

4.04 

4.04 

4.04 

4.03 

4.03 

4.03 

4.03 

4.02 

4.02 

4.02 

4.01 

4.01 

4.01 

4.01 

4.00 

4.00 

4.00 

3.99 

3.99 

3.99 

3.99 

3.98 

3.98 

3.98 

3.98 

3.97 

3.97 

3.97 

3.96 

3.96 

9.949881 

.949816 

.949752 

.949688 

.949623 

.949558 

.949494 

.949429 

.949364 

.949300 

9.949235 

.949170 

.949105 

.949040 

.948975 

.948910 

.948845 

.948780 

.948715 

.948650 

9.948584 

.948519 

.948454 

.948388 

.948323 

.948257 

.948192 

.948126 

.948060 

.947995 

9.947929 

.947863 

.947797 

.947731 

.947665 

.947600 

.947533 

.947467 

.947401 

.947335 

9.947269 

.947203 

.947136 

.947070 

.947004 

.946937 

.946871 

.946804 

.946738 

.946671 

9.946604 

.946538 

.946471 

.946404 

.946337 

.946270 

.946203 

.946136 

.946069 

.946002 

.945935 

1.07 

1.07 

1.07 

1.08 

1.08 

1.08 

1.08 

1.08 

1.08 

1.08 

1.08 

1.08 

1.08 

1 08 
1.08 
1.08 
1.09 
1.09 
1.09 
1.09 

1.09 

1 09 
1.09 
1.09 
1.09 
1.09 
1.09 
1.09 
1.09 
1.10 

1.10 

1.10 

1.10 

1.10 

1.10 

1.10 

1.10 

1.10 

1.10 

1.10 

1.10 

1.11 

1.11 

1.11 

1.11 

1.11 

1.11 

1.11 

1.11 

1.11 

1.11 

1.11 

1.11 

1.11 

1.12 

1.12 

1.12 

1.12 

1.12 

1.12 

9.707166 

.707478 

.707790 

.708102 

.708414 

.708726 

.709037 

.709349 

.709660 

.709971 

9.710282 

.710593 

.710904 

.711215 

.711525 

.711836 

.712146 

.712456 

.712766 

.713076 

9.713386 

.713696 

.714005 

.714314 

.714624 

.714933 

.715242 

.715551 

.715860 

.716168 

9.716477 

.716785 

.717093 

.717401 

.717709 

.718017 

.718325 

.718633 

.718940 

.719248 

9.719555 

.719862 

.720169 

.720476 

.720783 

.721089 

.721396 

.721702 

.722009 

.722315 

9.722621 

.722927 

.723232 

.723538 

.723844 

.724149 

.724454 

.724760 

.725065 

.725370 

.725674 

5.20 

5.20 

5.20 

5.20 

5.20 

5.19 

5.19 

5.19 

5.19 

5.18 

5.18 

5.18 

5.18 

5.18 

5.17 

5.17 

5.17 

5.17 

5.17 

5.16 

5.16 

5.16 

5.16 

5.15 

5.15 

5.15 

5.15 

5.15 

5.14 

5.14 

5.14 

5.14 

5.14 

5.13 

5.13 

5.13 

5.13 

5.13 

5.12 

5.12 

5.12 

5.12 

5.11 

5.11 

5.11 

5.11 

5.11 

5.10 

5.10 

5.10 

5.10 

5.10 

5.09 

5.09 

5.09 

5.09 

5.09 

5.08 

5.08 

5.08 

10.292834 

.292522 

.292210 

.291898 

.291586 

.291274 

.290963 

.290651 

.290340 

.290029 

10.289718 

.289407 

.289096 

.288785 

.288475 

.288164 

.287854 

.287544 

.287234 

.286924 

10.286614 

.286304 

.285995 

.285686 

.285376 

.285067 

.284758 

.284449 

.284140 

.283832 

10.283523 
.283215 
.282907 
.282599 
.282291 
.281983 
.281675 
.281367 
.281060 
.280752 

10.280445 

.280138 

.279831 

.279524 

.279217 

.278911 

.278604 

.278298 

.277991 

.277685 

10.277379 

.277073 

.276768 

.276462 

.276156 

.275851 

.275546 

.275240 

.274935 

.274630 

.274326 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 - 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 

M. 

Cosine. 

D.l". 

Sine. 

D.l". 

Co tang. 

D.l". 

Tang. 

M. 


117 


03 “ 

































TABLE IV. LOGARITHMIC SINES, ETC 


68 


28 ° 


M. 

Sine. 

D. 1". 

Cosine. 

D. 1". 

Tang. 

D. 1". 

Cotang. 

M. 

0 

1 

2 

3 

4 

5 

G 

7 

8 

9 

10 

11 

12 

13 

14 

15 
1G 

17 

18 

19 

20 
21 
22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 
3G 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 

9.671609 

.671847 

.672084 

.672321 

.672558 

.672795 

.673032 

.673268 

.673505 

.673741 

9.673977 

.674213 

.674448 

.674684 

.674919 

.675155 

.675390 

.675624 

.675859 

.676094 

9.676328 

.676562 

.676796 

.677030 

.677264 

.677498 

.677731 

.677964 

.678197 

.678430 

9.678663 

.678895 

.679128 

.679360 

.679592 

.679824 

.680056 

.680288 

.680519 

.680750 

9.680982 

.681213 

.681443 

.681674 

.681905 

.682135 

.682365 

.682595 

.682825 

.683055 

9.683284 

.683514 

.683743 

.683972 

.684201 

.684430 

.684658 

.684887 

.685115 

.685343 

.685571 

3.96 

3.96 

3.95 

3.95 

3.95 

3.94 

3.94 

3.94 

3.94 

3.93 

3.93 

3.93 

3.93 

3.92 

3.92 

3.92 

3.91 

3.91 

3.91 

3.91 

3.90 

3.90 

3.90 

3.90 

3.89 

3.89 

3.89 

3.88 

3.88 

3.88 

3.88 

3.87 

3.87 

3.87 

3.87 

3.86 

3.86 

3.86 

3.86 

3.85 

3.85 

3.85 

3.84 

3.84 

3.84 

3.84 

3.83 

3.83 

3.83 

3.83 

3.82 

3.82 

3.82 

3.82 

3.81 

3.81 

3.81 

3.80 

3.80 

3.80 

9.945935 

.945868 

.945800 

.945733 

.945666 

.945598 

.945531 

.945464 

.945396 

.945328 

9.945261 

.945193 

.945125 

.945058 

.944990 

.944922 

.944854 

.944786 

.944718 

.944650 

9.944582 

.944514 

.944446 

.944377 

.944309 

.944241 

.944172 

.944104 

.944036 

.943967 

9.943899 

.94:3830 

.943761 

.943693 

.943624 

.943555 

.943486 

.943417 

.943348 

.943279 

9.943210 
.943141 
.943072 
.943003 
.942934 
.942864 
.942795 
.942726 
.942656 
.942587 

9.942517 

.942448 

.942378 

.942308 

.942239 

.942169 

.942099 

.942029 

.941959 

.941889 

.941819 

1.12 

1.12 

1.12 

1.12 

1.12 

1.12 

1.12 

1.13 

1.13 

1.13 

1.13 

1.13 

1.13 

1.13 

1.13 

1.13 

1.13 

1.13 

1.13 

1.13 

1.14 
1.14 
1.14 
1.14 
1.14 
1.14 
1.14 
1.14 
1.14 
1.14 

1.14 

1.14 

1.15 
1.15 
1.15 
1.15 
1.15 
1.15 
1.15 
1.15 

1.15 

1.15 

1.15 

1.15 

1.15 

1.16 
1.16 
1.16 
1.16 
1.16 

1.16 

1.16 

1.16 

1.16 

1.16 

1.16 

1.16 

1.17 

1.17 

1.17 

9.725674 

.725979 

.726284 

.726588 

.726892 

.727197 

.727501 

.727805 

.728109 

.728412 

9.728716 

.729020 

.729323 

.729626 

.729929 

.730233 

.730535 

.730838 

.731141 

.731444 

9.731746 

.732048 

.732351 

.732653 

.732955 

.733257 

.733558 

.733860 

.734162 

.734463 

9.734764 

.735066 

.735367 

.735668 

.735969 

.736269 

.736570 

.736871 

.737171 

.737471 

9.737771 

.738071 

.738371 

738671 

.738971 

.739271 

.739570 

.739870 

.740169 

.740468 

9.740767 

.741066 

.741365 

.741664 

.741962 

.742261 

.742559 

.742858 

.743156 

.743454 

.743752 

5.08 

5.08 

5.07 

5.07 

5.07 

5.07 

5.07 

5.06 

5.06 

5.06 

5.06 

5.06 

5.05 

5.05 

5.05 

5.05 

5.05 

5.05 

5.04 

5.04 

5.04 

5.04 

5.04 

5.04 

5.03 

5.03 

5.03 

5.03 

5.02 

5.02 

5.02 

5.02 

5.02 

5.01 

5.01 

5.01 

5.01 

5.01 

5.01 

5.00 

5.00 

5.00 

5.00 

5.00 

4.99 

4.99 

4.99 

4.99 

4.99 

4.98 

4.98 

4.98 

4.98 

4.98 

4.98 

4.97 

4.97 

4.97 

4.97 

4.97 

10.274326 

.274021 

.273716 

.273412 

.273108 

.272803 

.272499 

.272195 

.271891 

.271588 

10.271284 
.270980 
.270677 
.270374 
.270071 
.269767 
.269465 
.269162 
.268859 
.268556 

10.268254 

.267952 

.267649 

.267347 

.267045 

.266743 

.266442 

.266140 

.265838 

.265537 

10.265236 
.264934 
.264633 
.264332 
.264031 
.263731 
.263430 
.263129 
.262829 
.262529 

10.262229 

.261929 

.261629 

.261329 

.261029 

.260729 

.2604:30 

.260130 

.259831 

.259532 

10.259233 

.258934 

.258635 

.258336 

.258038 

.257739 

.257441 

.257142 

.256844 

.256546 

.256248 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 

M. 

Cosine. 

D. 1". 

Sine. 

D. 1 

Cotang. 

D. 1". 

Tang. • 

M. 







































29 


TABLE IY. LOGARITHMIC SINES, ETC, 


69 


150 * 


M. 

Sine. 

| D. 1”. 

Cosine. 

I). 1". 

Tang. 

D. 1”. 

Cotang. 

M. 

0 

1 

2 

3 

4 

5 
G 

7 

8 
9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 
21 
22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 

9.685571 

.685799 

.686027 

.686254 

.686482 

.686709 

.686936 

.687163 

.687389 

.687616 

9.687843 

.688069 

.688295 

.688521 

.688747 

.688972 

.689198 

.689423 

.689648 

.689873 

9.690098 

.690323 

.690548 

.690772 

.690996 

.691220 

.691444 

.691668 

.691892 

.692115 

9.692339 

.692562 

.692785 

.693008 

.693231 

.693453 

.693676 

.693898 

.694120 

.694342 

9.694564 

.694786 

.695007 

.695229 

.695450 

.695671 

.695892 

.696113 

.696334 

.696554 

9.696775 

.696995 

.697215 

.697435 

.697654 

.697874 

.698094 

.698313 

.698532 

.698751 

.698970 

3.80 

3.79 

3.79 

3.79 

3.79 

3.78 

3.78 

3.78 

3.78 

3.77 

3.77 

3.77 

3.77 

3.76 

3.76 

3.76 

3.76 

3.75 

3.75 

3.75 

3.75 

3.74 

3.74 

3.74 

3.74 

3.73 

3.73 

3.73 

3.73 

3.72 

3.72 

3.72 

3.72 

3.71 

3.71 

3.71 

3.71 

3.70 

3.70 

3.70 

3.70 

3.69 

3.69 

3.69 

3.69 

3.68 

3.68 

3.68 

3.68 

3.67 

3.67 

3.67 

3.67 
3.66 
3.66 
3.66 
3.66 
3.65 
3.65 
3.65 

9.941819 

.941749 

.941679 

.941609 

.941539 

.941469 

.941398 

.941328 

.941*58 

.941187 

9.941117 

.941046 

.940975 

.940905 

.940834 

.940763 

.940693 

.940622 

.940551 

.940480 

9.940409 

.940338 

.940267 

.940196 

.940125 

.940054 

.939982 

.939911 

.939840 

.939768 

9.939697 

.939625 

.939554 

.939482 

.939410 

.939339 

.939267 

.939195 

.939123 

.939052 

9.938980 

.938908 

.938836 

.938763 

.938691 

.938619 

.938547 

.938475 

.938402 

.938330 

9.938258 

.938185 

.938113 

.938040 

.937967 

.937895 

.937822 

.937749 

.937676 

.937604 

.937531 

1.17 

1.17 

1.17 

1.17 

1.17 

1.17 

1.17 

1.17 

1.17 

1.17 

1.18 
1.18 
1.18 
1.18 
1.18 
1.18 
1.18 
1.18 
1.18 
1.18 

1.18 

1.18 

1.19 

1.19 

1.19 

1.19 

1.19 

1.19 

1.19 

1.19 

1.19 

1.19 

1.19 

1.19 

1.19 

1.20 
1.20 
1.20 
1.20 
1.20 

1.20 

1.20 

1.20 

1.20 

1.20 

1.20 

1.20 

1.21 

1.21 

1.21 

1.21 

1.21 

1.21 

1.21 

1.21 

1.21 

1.21 

1.21 

1.21 

1.22 

9.743752 

.744050 

.744348 

.744645 

.744943 

.745240 

.745538 

.745835 

.746132 

.746429 

9.746726 

.747023 

.747319 

.747616 

.747913 

.748209 

.748505 

.748801 

.749097 

.749393 

9.749689 

.749985 

.750281 

.750576 

.750872 

.751167 

.751462 

.751757 

.752052 

.752347 

9.752642 

.752937 

.753231 

.753526 

.753820 

.754115 

.754409 

.754703 

.754997 

.755291 

9.755585 

.755878 

.756172 

.756465 

.756759 

.757052 

.757345 

.757638 

.757931 

.758224 

9.758517 

.758810 

.759102 

.759395 

.759687 

.759979 

.760272 

.760564 

.760856 

.761148 

.761439 

4.96 

4.96 

4.96 

4.96 

4.96 

4.96 

4.95 

4.95 

4.95 

4.95 

4.95 

4.95 

4.94 

4.94 

4.94 

4.94 

4.94 

4.93 

4.93 

4.93 

4.93 

4.93 

4.93 

4.92 

4.92 

4.92 

4.92 

4.92 

4.92 

4.91 

4.91 

4.91 

4.91 

4.91 

4.91 

4.90 

4.90 

4.90 

4.90 

4.90 

4.89 

4.89 

4.89 

4.89 

4.89 

4.89 

4.88 

4.88 

4.88 

4.88 

4.88 

4.88 

4.87 

4.87 

4.87 

4.87 

4.87 

4.87 

4.86 

4.86 

10.256248 

.255950 

.255652 

.255355 

.255057 

.254760 

.254462 

.254165 

.253868 

.253571 

10.253274 

.252977 

.252681 

.252384 

.252087 

.251791 

.251495 

.251199 

.250903 

.250607 

10.250311 

.250015 

.249719 

.249424 

.249128 

.248833 

.248538 

.248243 

.247948 

.247653 

10.247358 

.247063 

.246769 

.246474 

.246180 

.245885 

.245591 

.245297 

.245003 

.244709 

10.244415 

.244122 

.243828 

.243535 

.243241 

.242948 

.242655 

.242362 

.242069 

.241776 

10.241483 

.241190 

.240898 

.240605 

.240313 

.240021 

.239728 

.239436 

.239144 

.238852 

.238561 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 

M. 

Cosine. 

D. 1 '. 

Sine. 

D.l”. 

Cotang. 

D. 1 ". 

Tang. 

M. 

















































TABLE IV. LOGARITHMIC BINES, ETC 


1.4tt 


70 


30 ° 


M. 

Sine. 

D. 1 

Cosine. 

D. 1". 

Tang. 

D. 1”. 

Cotang. 

M. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 
1G 

17 

18 

19 

20 
21 
22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 
83 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 
63 

54 

55 
66 

67 

68 

59 

60 

9.698970 

.699189 

.699407 

.699626 

.699844 

.700062 

.700280 

.700498 

.700716 

.700933 

9.701151 

.701368 

.701585 

.701802 

.702019 

.702236 

.702452 

.702669 

.702885 

.703101 

9.703317 

.703533 

.703749 

.703964 

.704179 

.704395 

.704610 

.704825 

.705040 

.705254 

9.705469 

.705683 

.705898 

.706112 

.706326 

.706539 

.706753 

.706967 

.707180 

.707393 

9.707606 

.707819 

.708032 

.708245 

.708458 

.708670 

.708882 

.709094 

.709306 

.709518 

9.709730 

.709941 

.710153 

.710364 

.710575 

.710786 

.710997 

.711208 

.711419 

.711629 

.711839 

3.65 

3.64 

3.64 

3.64 

3.64 

3.63 

3.63 

3.63 

3.63 

3.62 

3.62 

3.62 

3.62 

3.61 

3.61 

3.61 

3.61 

3.60 

3.60 

3.60 

3.60 

3.59 

3.59 

3.59 

3.59 

3.59 

3.58 

3.58 

3.58 

3.58 

3.57 

3.57 

3.57 
3.57 
3.56 
3.56 
3.56 
3.56 
3.55 
3.55 

3.55 

3.55 

3.54 

3.54 

3.54 

3.54 

3.54 

3.53 

3.53 

3.53 

3.53 

3.52 

3.52 

3.52 

3.52 

3.51 

3.51 

3.51 

3.51 

3.51 

9.937531 

.937458 

.937385 

.937312 

.937238 

.937165 

.937092 

.937019 

.936946 

.936872 

9.936799 

.936725 

.936652 

.936578 

.936505 

.936431 

.936357 

.936284 

.936210 

.936136 

9.936062 

.935988 

.935914 

.935840 

.935766 

.935692 

.935618 

.935543 

.935469 

.935395 

9.935320 

.935246 

.935171 

.935097 

.935022 

.934948 

.934873 

.934798 

.934723 

.934649 

9.934574 

.934499 

.934424 

.934349 

.934274 

.934199 

.934123 

.934048 

.933973 

.933898 

9.933822 

.933747 

.933671 

.933596 

.933520 

.933445 

.933369 

.933293 

.933217 

.933141 

.933066 

1.22 

1.22 

1.22 

1.22 

1.22 

1.22 

1.22 

1.22 

1.22 

1.22 

1.22 

1.23 

1.23 

1.23 

1.23 

1.23 

1.23 

1.23 

1.23 

1.23 

1.23 

1.23 

1.23 

1.23 

1.24 
1.24 
1.24 
1.24 
1.24 
1.24 

1.24 

1.24 

1.24 

1.24 
1.24 

1.24 

1.25 
1.25 
1.25 
1.25 

1.25 

1.25 

1.25 

1.25 

1.25 

1.25 

1.25 

1.25 

1.26 
1.26 

1.26 

1.26 

1.26 

1.26 

1.26 

1.26 

1.26 

1.26 

1.26 

1.26 

9.761439 

.761731 

.762023 

.762314 

.762606 

.762897 

.763188 

.763479 

.763770 

.764061 

9.764352 

.764643 

.764933 

.765224 

.765514 

.765805 

.766095 

.766385 

.766675 

.766965 

9.767255 

.767545 

.767834 

.768124 

.768413 

.768703 

.768992 

.769281 

.769571 

.769860 

9.770148 

.770437 

.770726 

.771015 

.771303 

.771592 

.771880 

.772168 

.772457 

.772745 

9.773033 

.773321 

.773608 

.773896 

.774184 

.774471 

.774759 

.775046 

.775333 

.775621 

9.775908 

.776195 

.776482 

.776769 

.777055 

.777342 

.777628 

.777915 

.778201 

.778488 

.778774 

4.86 

4.86 

4.86 

4.86 

4.86 

4.85 

4.85 

4.85 

4.85 

4.85 

4.85 

4.84 

4.84 

4.84 

4.84 

4.84 

4.84 

4.83 

4.83 

4.83 

4.83 

4.83 

4.83 

4.82 

4.82 

4.82 

4.82 

4.82 

4.82 

4.82 

4.81 

4.81 

4.81 

4.81 

4.81 

4.81 

4.80 

4.80 

4.80 

4.80 

4.80 

4.80 

4.80 

4.79 

4.79 

4.79 

4.79 

4.79 

4.79 

4.78 

4.78 

4.78 

4.78 

4.78 

4.78 

4.78 

4.77 

4.77 

4.77 

4.77 

10.238561 

.238269 

.237977 

.237686 

.237394 

.237103 

.236812 

.236521 

.236230 

.235939 

10.235648 

.235357 

.235067 

.234776 

.234486 

.234195 

.233905 

.233615 

.233325 

.233035 

10.232745 

.232455 

.232166 

.231876 

.231587 

.231297 

.231008 

.230719 

.230429 

.230140 

10.229852 

.229563 

.229274 

.228985 

.228697 

.228408 

.228120 

.227832 

.227543 

.227255 

10.226967 

.226679 

.226392 

.226104 

.225816 

.225529 

.225241 

.224954 

.224667 

.224379 

10.224092 

.223805 

.223518 

.223231 

.222945 

.222658 

.222372 

.222085 

.221799 

.221512 

.221226 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 

M. 

Cosine. 

D. 1 

Sine. 

D. 1”. 

Cotang. 

D. 1". 

Tang. 

M. 


ISO 


59 










































31 


TABLE IV. LOGARITHMIC SINES, ETC. 


71 


148 ° 


M. 

Sine. 

D.l". 

Cosine. 

D.l" 

Tung. 

D.l". 

Cotang. 

M. 

0 

1 

2 

3 

4 

5 
G 

7 

8 
9 

10 

11 

12 

13 

14 

15 
1G 

17 

18 

19 

20 
21 
22 

23 

24 

25 
2G 

27 

28 

29 

30 

31 

32 

33 

34 

35 
3G 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 
69 
60 

9.711839 

.712050 

.7122G0 

.712469 

.712679 

.712889 

.713098 

.713308 

.713517 

.713726 

9.713935 

.714144 

.714352 

.714561 

.714769 

.714978 

.715186 

.715394 

.715602 

.715809 

9.716017 

.716224 

.716432 

.716639 

.716846 

.717053 

.717259 

.717466 

.717673 

.717879 

9.718085 

.718291 

.718497 

.718703 

.718909 

.719114 

.719320 

.719525 

.719730 

.719935 

9.720140 

720345 

.720549 

.720754 

.720958 

.721162 

.721366 

.721570 

.721774 

.721978 

9.722181 

.722385 

.722588 

.722791 

.722994 

.723197 

.723400 

.723603 

.723805 

.724007 

.724210 

3..50 
3.50 
3.50 
3.50 
3.49 
3.49 
3.49 
3.49 
3.48 
3.48 

3.48 

3.48 

3.48 

3.47 

3.47 

3.47 

3.47 

3.46 

3.46 

3.46 

3.46 

3.46 

3.45 

3.45 

3.45 

3.45 

3.44 

3.44 

3.44 

3.44 

3.43 

3.43 

3.43 

3.43 

3.43 

3.42 

3.42 

3.42 

3.42 

3.41 

3.41 

3.41 

3.41 

3.41 

3.40 

3.40 

3.40 

3.40 

3.39 

3.39 

3.39 

3.39 

3.39 

3.38 

3.38 

3.38 

3.38 

3.37 

3.37 

3.37 

9.933066 

.932990 

.932914 

.932838 

.932762 

.932685 

932609 

.932533 

.932457 

.932380 

9.932304 

.932228 

.932151 

.932075 

.931998 

.931921 

.931845 

.931768 

.931691 

.931614 

9.931537 

.931460 

.931383 

.931306 

.931229 

.931152 

.931075 

.930998 

.930921 

.930843 

9.930766 
.930688 
.930611 
.930533 
.930456 
.930378 
.930300 
.930223 
.930145 
.930067 

9.929989 

.929911 

.929833 

.929755 

.929677 

.929599 

.929521 

.929442 

.929364 

.929286 

9.929207 

.929129 

.929050 

.928972 

.928893 

.928815 

.928736 

.928657 

.928578 

.928499 

.928420 

1.27 

1.27 

1.27 

1.27 

1.27 

1.27 

1.27 

1.27 

1.27 

1.27 

1.27 

1 27 

1.28 
1.28 
1.28 
1,28 
1.28 
1.28 
1.28 
1.28 

1.28 

1.28 

1.28 

1.28 

1.29 

1.29 

129 

1.29 

1.29 

1.29 

1.29 

1.29 

1.29 

1 29 
1.29 

1.29 

1.30 
1.30 
1.30 
1.30 

1.30 

1.30 

1.30 

1.30 

1.30 

1.30 

1.30 

1.31 
1.31 
1.31 

1.31 

1.31 

1.31 

1.31 

1.31 

1.31 

1.31 

1.31 

1.31 

1.32 

9.778774 

.779060 

.779346 

.779632 

.779918 

.780203 

.780489 

.780775 

.781060 

.781346 

9.781631 

.781916 

.782201 

.782486 

.782771 

.783056 

.783341 

.783626 

.783910 

.784195 

9.784479 

.784764 

.785048 

.785332 

.785616 

.785900 

.786184 

.786468 

.786752 

.787036 

9.787319 

.787603 

.787886 

.788170 

.788453 

.788736 

.789019 

.789302 

.789585 

.789868 

9.790151 

.790433 

.790716 

.790999 

.791281 

.791563 

.791846 

.792128 

.792410 

.792692 

9.792974 

.793256 

.793538 

.793819 

.794101 

.794383 

.794664 

.794945 

.795227 

.795508 

.795789 

4.77 
4.77 
4.77 
4.76 
4 76 
4.76 
4 76 
4.76 
4.76 
4.76 

4.75 

4.75 

4.75 

4.75 

4.75 

4.75 

4.75 

4.74 

4.74 

4.74 

4.74 

4.74 

4.74 

4.74 

4 73 

4 73 
4.73 
4.73 
4.73 
4.73 

4.73 

4.72 

4.72 

4.72 

4.72 

4.72 

4.72 

4.72 

4.71 

4.71 

4.71 

4.71 

4.71 

4.71 

4.71 

4.70 

4.70 

4.70 

4.70 

4.70 

4.70 

4.70 

4.70 

4.69 \ 

4.69 

4.69 

4.69 

4.69 

4.69 

4.69 

10.221226 

.220940 

.220654 

.220368 

.220082 

.219797 

.219511 

.219225 

.218940 

.218654 

10.218369 

.218084 

.217799 

.217514 

.217229 

.216944 

.216659 

.216374 

.216090 

.215805 

10.215521 

.215236 

.214952 

.214668 

.214384 

.214100 

.213816 

.213532 

.213248 

.212964 

10.212681 

.212397 

.212114 

.211830 

.211547 

.211264 

.210981 

.210698 

.210415 

.210132 

10.209849 

.209567 

.209284 

.209001 

.208719 

.208437 

.208154 

.207872 

.207590 

.207308 

10.207026 

.206744 

.206462 

.206181 

.205899 

.205617 

.205336 

.205055 

.204773 

.204492 

.204211 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 r 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 ' 

6 

5 

4 

3 

2 

1 

0 

M. 

Cosine. 

D.l". 

Sine. 

D.l". 

Cotang. 

D.l',’. 

Tang. 

M. 





































































72 TABLE IV. LOGARITHMIC SINES, ETC. 


M. 

Sine. 

D.l". 

Cosine. 

D. 1 '. 

Tang. 

D. 1". 

Cotang. 

M. 

0 

.1 

2 

3 

4 

5 

G 

7 

8 

9 

10 

11 

12 

13 

14 

15 
1G 

17 

18 

19 

20 
21 

22 

23 

24 

25 
2G 

27 

28 

29 

30 

31 

32 

33 

34 

35 
3G 

37 

38 

39 

40 

41 

42 

43 

44 

45 
4G 

47 

48 

49 

50 

51 

52 

53 

54 

55 
5G 

57 

58 

59 

60 

9.724210 

.724412 

.724614 

.72481G 

.725017 

.725219 

.725420 

.725622 

.725823 

.726024 

9.726225 

.726426 

.726626 

.726827 

.727027 

.727228 

.727428 

.727628 

.727828 

.728027 

9.728227 

.728427 

.728626 

.728825 

.729024 

.729223 

.729422 

.729621 

.729820 

.730018 

9.730216 

.730415 

.730613 

.730811 

.731009 

.731206 

.731404 

.731602 

.731799 

.731996 

9.732193 

.732390 

.732587 

.732784 

.732980 

.733177 

.733373 

.733569 

.733765 

.733961 

9.734157 

.734353 

.734549 

.734744 

.734939 

.735135 

.735330 

.735525 

.735719 

.735914 

.736109 

3.37 

3.37 

3.36 

3.36 

3.36 

3.36 

3.36 

3.35 

3.35 

3.35 

3.35 

3.34 

3.34 

3.34 

3.34 

3.34 

3.33 

3.33 

3.33 

3.33 

3.33 

3.32 

3.32 

3.32 

3.32 

3.31 

3.31 

3.31 

3.31 

3.31 

3.30 

3.30 

3.30 

3.30 

3.30 

3.29 

3.29 

3.29 

3.29 

3.28 

3.28 

3.28 

3.28 

3.28 

3.27 

3.27 

3.27 

3.27 

3.27 

3.26 

3 26 
3.26 
3.26 
3.26 
3.25 
3.25 
3.25 
3.25 
3.25 
3.24 

9.928420 

.928342 

.928263 

.928183 

.928104 

.928025 

.927946 

.927867 

.927787 

.927708 

9.927629 

.927549 

.927470 

.927390 

.927310 

.927231 

.927151 

.927071 

.926991 

.926911 

9.926831 

.926751 

.926671 

.926591 

.926511 

.926431 

.926351 

.926270 

.926190 

.926110 

9.926029 

.925949 

.925868 

.925788 

.925707 

.925626 

.925545 

.925465 

.925384 

.925303 

9.925222 
.925141 
.925060 
* .924979 
.924897 
.924816 
.924735 
.924654 
.924572 
.924491 

9.924409 

.924328 

.924246 

.924164 

.924083 

.924001 

.923919 

.923837 

.923755 

.923673 

.923591 

1.32 

1.32 

1.32 

1.32 

1.32 

1.32 

1.32 

1.32 

1.32 

1.32 

1.32 

1.33 
1.33 
1.33 
1.33 
1.33 
1.33 
1.33 
1.33 
1.33 

1.33 

1.33 

1.33 

1.34 
1.34 
1.34 
1.34 
1.34 
1.34 
1.34 

1.34 

1.34 

1.34 

1.34 

1.35 
1.35 
1.35 
1.35 
1.35 
1.35 

1.35 

1.35 

1.35 

1.35 

1.35 

1.35 

1.36 
1.36 
1.36 
1.36 

1.36 

1.36 

1.36 

1.36 

1.36 

1.36 

1.36 

1.37 
1.37 
1.37 

9.795789 

.796070 

.796351 

.796632 

.796913 

.797194 

.797475 

.797755 

.798036 

.798316 

9.798596 
.798877 
.799157 
.799437 
.799717 
.799997 
.800277 
.800557 
.800836 
.801116 

9.801396 

.801675 

.801955 

.802234 

.802513 

.802792 

.803072 

.803351 

.803630 

.803908 

9.804187 

.804466 

.804745 

.805023 

.805302 

.805580 

.805859 

.806137 

.806415 

.806693 

9.806971 

.807249 

.807527 

.807805 

.808083 

.808361 

.808638 

808916 

.809193 

.809471 

9.809748 

.810025 

.810302 

.810580 

.810857 

.811134 

.811410 

.811687 

.811964 

.812241 

.812517 

4.68 

4.68 

4.68 

4.68 

4.68 
4.68 
4.68 
4.68 
4.67 
4.67 

4.67 

4.67 

4.67 

4.67 

4.67 

4.66 

4.66 

4.66 

4.66 

4.66 

4.66 

4.66 

4.66 

4 65 
'4.65 
4.65 
4.65 
4.65 
4.65 
4.65 

4.65 

4.64 

4.64 

4.64 

4.64 

4.64 

4.64 

4.64 

4.64 

4.63 

4.63 

4.63 

4.63 

4.63 

4.63 

4.63 

4.63 

4.62 

4.62 

4.62 

4.62 

4.62 

4.62 

4.62 

4.62 

4.61 

4.61 

4.61 

4.61 

4.61 

10.204211 

.203930 

.203649 

.203368 

.203087 

.202806 

.202525 

.202245 

.201964 

.201684 

10.201404 
.201123 
.200843 
.200563 
.200283 
.200003 
.199723 
.199443 
.199164 
.198884 

10.198604 
.198325 
.198045 
.197766 
.1974S7 
.197208 
.196928 
.196649 
.196370 
.196092 

10.195313 
.195534 
.195255 
.194977 
.194698 
.194420 
.194141 
.193863 
.193585 
.193307 

10.193029 

.192751 

.192473 

.192195 

.191917 

.191639 

.191362 

.191084 

.190807 

.190529 

10.190252 

.189975 

.189698 

.189420 

.189143 

.188866 

.188590 

.188313 

.188036 

.187759 

.187483 

GO 

59 

58 

67 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 , 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 

M 

1 Cosine. 

D.l'. 

Sine. 

D.l". 

Cotang. 

D.l". 

Tang. 

M. 


57 ° 


122 








































TABLE IV. LOGARITHMIC SINES, ETC. 73 


S3 0 146° 


M. 

Sine. 

D.l”. 

Cosine. 

D.l" 

Tang 

D.l’’. 

Cotang. 

M. 

0 

1 

2 

3 

4 

5 

G 

7 

8 

D 

30 

11 

12 

13 

14 

15 
1G 

17 

18 

19 

20 
21 
22 

23 

24 

25 
2G 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 
4G 

47 

48 

49 

50 

51 

52 

53 

54 

55 
5G 

57 

58 

59 
GO 

9.73G109 

.73G303 

.736498 

.736692 

.736886 

.737080 

.737274 

.737467 

.737661 

.737855 

9.738048 

.738241 

.738434 

.738627 

.738820 

.739013 

.739206 

.739398 

.739590 

.739783 

9.739975 

.740167 

.740359 

.740550 

.740742 

.740934 

.74112;) 

.741316 

.741508 

.741699 

9.741889 

.742080 

.742271 

.742462 

.742652 

.742842 

.743033 

.743223 

.743413 

.743602 

9.743792 

.743982 

.744171 

.744361 

.744550 

.744739 

.744928 

.745117 

.745306 

.745494 

9.745683 

.745871 

.746060 

.746248 

.746436 

.746624 

.746812 

.746999 

.747187 

.747374 

.747562 

3.24 

3.24 

3.24 

3.23 

3.23 

3.23 

3.23 

3.23 

3.22 

3.22 

3.22 

3.22 

3.22 

3.21 

3.21 

3.21 

3.21 

3.21 

3.20 

3.20 

3.20 

3.20 

3.20 

3.19 

3.19 

3.19 

3.19 

3.19 

3.18 

3.18 

3.18 

3.18 

3.18 

3.17 

3.17 

3.17 

3.17 

3.17 

3.16 

3.16 

3.16 

3.16 

3.16 

3.15 

3.15 

3.15 

3.15 

3.15 

3.14 

3.14 

3.14 

3.14 

3.14 

3.13 

3.13 

3.13 

3.13 

3.13 

3.12 

3.12 

9.923591 

.923509 

.923427 

.923345 

.923263 

.923181 

.923098 

.923016 

.922933 

.922851 

9.922768 

.922686 

.922603 

.922520 

.922438 

.922355 

.922272 

.922189 

.922106 

.922023 

9.921940 

.921857 

.921774 

.921691 

.921607 

.921524 

.921441 

.921357 

.921274 

.921190 

9.921107 

.921023 

.920939 

.920856 

.920772 

.920688 

.920604 

.920520 

.920436 

.920352 

9.920268 

.920184 

.920099 

.920015 

.919931 

.919846 

.919762 

.919677 

.919593 

.919508 

9.919424 

.919339 

.919254 

.919169 

.919085 

.919000 

.918915 

.918830 

.918745 

.918659 

.918574 

1.37 

1.37 

1.37 

1.37 

1.37 

1.37 

1.37 

1.37 
1.37. 

1.38 

1.38 

1.38 

1.38 

1.38 

1.38 

1.38 

1.38 

1.38 

1.38 

1.38 

1.39 
1.39 
1.39 
1.39 
1.39 
1.39 
1.39 
1.39 
1.39 
1.39 

1.39 

1.39 

1.40 
1.40 
1.40 
1.40 
1.40 
1.40 
1.40 
1.40 

1.40 

1.40 

1.40 

1.41 
1.41 
1.41 
1.41 
1.41 
1.41 
1.41 

1.41 

1.41 

1.41 

1.41 

1.42 
1.42 
1.42 
1.42 
1.42 
1.42 

9.812517 

.812794 

.813070 

.813347 

.813623 

.813899 

.814176 

.814452 

.814728 

.815004 

9.815280 

.815555 

.815831 

.816107 

.816382 

.816658 

.816933 

.817209 

.817484 

.817759 

9.818035 

.818310 

.818585 

.818860 

.819135 

.819410 

.819684 

.819959 

.820234 

.820508 

9.820783 

.821057 

.821332 

.821606 

.821880 

.822154 

.822429 

.822703 

.822977 

.823250 

9.823524 

.823798 

.824072 

.824345 

.824619 

.824893 

.825166 

.825439 

.825713 

.825986 

9.826259 

.826532 

.826805 

.827078 

.827351 

.827624 

.827897 

.828170 

.828442 

.828715 

.828987 

4.61 

4.61 

4.61 

4.61 

4.60 

4.60 

4.60 

4.60 

4.60 

4.60 

4.60 

4.60 

4.59 

4.59 

4.59 

4.59 

4.59 

4.59 

4.59 

4.59 

4.59 
4.58 
4.58 
4.58 
4-. 58 
4.58 
4.58 
4.58 
4.58 
4.58 

4.57 

4.57 

4.57 

4.57 

4.57 

4.57 

4.57 

4.57 

4.57 

4.56 

4.56 

4.56 

4.56 

4.56 

4.56 

4.56 

4.56 

4.56 

4.55 

4.55 

4.55 

4.55 

4.55 

4.55 

4.55 

4.55 

4.55 

4.54 

4.54 

4.54 

10.187483 
.187206 
.186930 
.186653 
.186377 
.186101 
.185824 
.185548 
.185272 
.184996 

10.184720 

.184445 

.184169 

.183893 

.183618 

.183342 

.183067 

.182791 

.182516 

.182241 

10.181965 

.181690 

.181415 

.181140 

.180865 

.180590 

.180316 

.180041 

.179766 

.179492 

10.179217 

.178943 

.178668 

.178394 

.178120 

.177846 

.177571 

.177297 

.177023 

.176750 

10.176476 

.176202 

.175928 

.175655 

.175381 

.175107 

.174834 

.174561 

.174287 

.174014 

10.173741 

.173468 

.173195 

.172922 

.172649 

.172376 

.172103 

.171830 

.171558 

.171285 

.171013 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

O 

M. | Cosine. 

D.l". 

Sine. 

D.l ’’. 

Co tang. 

D.l’'. 

Tang. 

M. 


123 . 


56 




































TABLE IV. LOGARITHMIC SINES, ETC, 


145 


74 


34 ° 


M. 

Sine. 

D.l". 

Cosine. 

I). l . 

Tang. 

D. 1". 

Cotang. 

M. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 
21 
22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 

9.747562 

.747749 

.747936 

.748123 

.748310 

.748497 

.748683 

.748870 

.749056 

.749243 

9.749429 

.719615 

.749801 

.749987 

.750172 

.750358 

.750543 

.750729 

.750914 

.751099 

9.751284 

.751469 

.751654 

.751839 

.752023 

.752208 

.752392 

.752576 

.752760 

.752944 

9.753128 

.753312 

.753495 

.753679 

.753862 

.754046 

.754229 

.754412 

.754595 

.754778 

9.754960 
.755143 
.755326 
.755508 
.755690 
.755872 
.756054 
.756236 
.756418 
.756600 

9.756782 

.756963 

.757144 

.757326 

.757507 

.757688 

.757869 

.758050 

.758230 

.758411 

.758591 

3.12 

3.12 

3.12 

3.11 

3.11 

3.11 

3.11 

3.11 

3.10 

3.10 

3.10 

3.10 

3.10 

3.10 

3.09 

3 09 
3.09 
3.09 
3.09 
3.08 

3.08 

3.08 

3.08 

3.08 

3.07 

3.07 

3.07 

3.07 

3.07 

3.06 

3.06 

3.06 

3.06 

3.06 

3.05 

3.05 

3.05 

3.05 

3.05 

3.05 

3.04 

3.04 

3.04 

3.04 

3.04 

3.03 

3.03 

3.03 

3.03 

3.03 

3.02 

3.02 

3.02 

3.02 

3.02 

3.02 

3.01 

3.01 

3.01 

3.01 

9.918574 

.918489 

.918404 

.918318 

.918233 

.918147 

.918062 

.917976 

.917891 

.917805 

9.917719 
.917634 
.917548 
.917462 
.917376 
.917290 
.917204 
.917118 
.917032 
.916946 

9.916859 

.916773 

.916687 

.916600 

.916514 

.916427 

.916341 

.916254 

.916167 

.916081 

9.915994 

.915907 

915820 

.915733 

.915646 

.915559 

.915472 

.915385 

.915297 

.915210 

9.915123 

.915035 

.914948 

.914860 

.914773 

.914685 

.914598 

.914510 

.914422 

.914334 

9.914246 

.914158 

.914070 

.913982 

.913894 

.913806 

.913718 

.913630 

.913541 

.913453 

.913365 

1.42 

1 42 
1.42 
1.42 

1.42 

1.43 
1.43 
1.4.3 
1.43 
1.43 

1.43 

1.43 
1.43 
1.43 
1.43 
1.43 

1.43 
1.41 

1.41 

1.44 

1.44 

1 41 
1.44 
1.44 

1.44 
1.44 

1 44 

1.44 
1.15 

1.45 

1.4!> 

1.45 

1.45 
1.45 

1.45 
1.45 
1.45 
1.45 

1.45 

1.46 

1.46 

1.46 

1.46 

1.46 

1.46 

1.46 

1.46 

1.46 

1.46 

1.46 

1 47 

1.47 
1.47 

1 47 
1.47 

1.47 
1.47 
1.47 

1.47 
1.47 

9.828987 
.829260 
.829532 
.829805 
.830077 
.830349 
.830621 
.830893 
.831165 
.831437 

9.831709 

.831981 

.832253 

.832525 

.832796 

.833068 

.833339 

.833611 

.833882 

.834154 

9.834425 

.834696 

.834967 

.835238 

.835509 

.835780 

.836051 

.836322 

.836593 

.836864 

9.837131 
.837405 
.837675 
.837946 
.838216 
.838487 
.838757 
.839027 
.839297 
.839568 

9.839838 

.840108 

.840378 

.840648 

.840917 

.841187 

.841457 

.841727 

.841996 

.842266 

9.842535 

.842805 

.843074 

.843343 

.843612 

.843882 

.844151 

.844420 

.841689 

.844958 

.845227 

4.54 

4.54 

4.54 

4.54 

4.54 

4.54 

4.53 

4.53 

4.53 

4.53 

4.53 

4.53 

4.53 

4.53 

4.53 

4.53 

4 52 
4.52 
4.52 
4.52 

4.52 

4.52 

4.52 

4.52 

4.52 

4.52 

4.51 

4.51 

4.51 

4.51 

4.51 

4.51 

4.51 

4.51 

4.51 

4.51 

4.50 

4.50 

4.50 

4.50 

4.50 

4.50 

4.50 

4.50 

4.50 

4.49 

4.49 

4.49 

4.49 

4.49 

4.49 

4.49 

4.49 

4.49 

4.49 

4.49 

4.48 

4.48 

4.48 

4.48 

10.171013 
.170740 
.170468 
.170195 
.169923 
.169651 
.169379 
.169107 
.168835 
.168563 

10.168291 
.168019 
.167747 
.167475 
.167204 
.166932 
.166661 
.166389 
.166118 
.165846 

10.165575 
.165304 
.165033 
.164762 
.164491 
.164220 
.163949 
.163678 
.163407 
.163136 

10.162866 
.162595 
.162325 
.162054 
.161784 
.161513 
.161243 
.160973 
.160703 
. 160132 

10 160162 
.159892 
.159622 
.159352 
.159083 
.158813 
.158543 
.158273 
.158004 
.157734 

10.157465 
.157195 
.156926 
.156657 
.156388 
.156118 
.155849 
.155580 
.155311 
.155652 
.154773 

60 

59 

58 

57 

66 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 

M. 

Cosine. 

D.l". 

Sine. 

D.l" 

Cotang. 

D. 1”. 

Tang. 

M. 





















































35 


TABLE IY. LOGARITHMIC SINES, ETC, 


75 


144 ° 


M. 

0 

1 

2 

3 

4 

5 

6 

7 

8 
9 

10 

11 

12 

13 

14 

15 
lf» 

17 

18 

19 

20 
21 
22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

62 

53 

64 

55 

66 

67 

68 

59 

60 

Sine. 

D.l". 

Cosine. 

D.l”. 

Tang. 

D.l”. 

Cotang. 

M. 

9.758591 

.758772 

.758952 

.759132 

.759312 

.759492 

.759672' 

.759852 

.760031 

.760211 

9.760390 

.760569 

.760748 

.760927 

.761106 

.761285 

.761464 

.761642 

.761821 

.761999 

9.762177 

.762356 

.762534 

.762712 

.762889 

.763067 

.763245 

.763422 

.763600 

.763777 

9.763954 

.764131 

.764308 

.764485 

.764662 

.764838 

.765015 

.765191 

.765367 

.765544 

9.765720 

.765896 

.766072 

.766247 

.766423 

.766598 

.766774 

.766949 

.767124 

.767300 

9.767475 

.767649 

.767824 

.767999 

.768173 

.768348 

.768522 

.768697 

.708871 

.769045 

.769219 

3.01 

3.00 

3.00 

3.00 

3.00 

3.00 

2.99 

2.99 

2.99 

2.99 

2.99 

2.99 

2.98 

2.98 

2.98 

2.98 

2.98 

2.97 

2.97 

2.97 

2.97 

2.97 

2.97 

2.96 

2.96 

2.96 

2.96 

2.96 

2.95 

2.95 

2.95 

2.95 

2.95 

2.95 

2.94 

2.94 

2.94 

2.94 

2.94 

2.93 

2.93 

2.93 

2.93 

2.93 

2.93 

2.92 

2.92 

2.92 

2.92 

2.92 

2.91 

2.91 

2.91 

2.91 

2.91 

2.91 

2.90 

2.90 

2.90 

2.90 

9.913365 

.913276 

.913187 

.913099 

.913010 

.912922 

.912833 

.912744 

.912655 

.912566 

9.912477 

.912388 

.912299 

.912210 

.912121 

.912031 

.911942 

.911853 

.911763 

.911674 

9.911584 

.911495 

.911405 

.911315 

.911226 

.911136 

.911046 

.910956 

.910866 

.910776 

9.910686 

.910596 

.910506 

.910415 

.910325 

.910235 

.910144 

.910054 

.909963 

.909873 

9.909782 

.909691 

.909601 

.909510 

.909419 

.909328 

.909237 

.909146 

.909055 

.908964 

9.908873 

.908781 

.908690 

.908599 

.908507 

.908416 

.908324 

.908233 

.908141 

.908049 

.907958 

1.47 

1.48 
1.48 
1.48 
1.48 
1.48 
1.48 
1.48 
1.48 
1.48 

1.48 

1.48 

1.49 
1.49 
1.49 
1.49 
1.49 
1.49 
1.49 
1.49 

1.49 

1.49 

1.49 

1.50 
1.50 
1.50 
1.50 
1.50 
1.50 
1.50 

1.50 

1.50 

1.50 

1.51 
1.51 
1.51 
1.51 
1.51 
1.51 
1.51 

1.51 

1.51 

1.51 

1.51 

1.52 
1.52 
1.52 
1.52 
1.52 
1.52 

1.52 

1.52 

1.52 

1.52 

1.52 

1.53 
1.53 
1.53 

1.53 
1.53 

9.845227 

.845496 

.845764 

.846033 

.846302 

.846570 

.846839 

.847108 

.847376 

.847644 

9.847913 

.848181 

.848449 

.848717 

.848986 

.849254 

.849522 

.849790 

.850057 

.850325 

9.850593 

.850861 

.851129 

.851396 

.851664 

.851931 

.852199 

.852466 

.852733 

.853001 

9.853268 

.853535 

.853802 

.854069 

.854336 

.854603 

.854870 

.855137 

.855404 

.855671 

9.855938 

.856204 

.856471 

.856737 

.857004 

.857270 

.857537 

.857803 

.858069 

.858336 

9.858602 

.858808 

.859134 

.859400 

.859666 

.859932 

.860198 

.860464 

.860730 

.860995 

.861261 

4.48 

4.48 

4.48 

4.48 

4.48 

4.48 

4.48 

4.47 

4.47 

4.47 

4.47 

4.47 

4.47 

4.47 

4.47 

4.47 

4.47 

4.46 

4.46 

4.46 

4.46 

4.46 

4.46 

4.46 

4.46 

4.46 

4.46 

4.46 

4.46 

4.45 

4.45 

4.45 

4.45 

4.45 

4.45 

4.45 

4.45 

4.45 

4.45 

4.44 

4.44 

4.44 

4.44 

4.44 

4.44 

4.44 

4.44 

4.44 

4.44 

4.44 

4.44 

4.43 

4.43 

4.43 

4.43 

4.43 

4.43 

4.43 

4.43 

4.43 

10.154773 

.154504 

.154236 

.153967 

.153698 

.153430 

.153161 

.152892 

.152624 

.152356 

10.152087 
.151819 
.151551 
.151283 
.151014 
.150746 
.150478 
.150210 
.149943 
.149675 

10.149407 

.149139 

.148871 

.148604 

.148336 

.148069 

.147801 

.147534 

.147267 

.146999 

10.146732 

.146465 

.146198 

.145931 

.145664 

.145397 

.145130 

.144863 

.144596 

.144329 

10.144062 

.143796 

.143529 

.143263 

.142996 

.142730 

.142463 

.142197 

.141931 

.141664 

10.141398 

.141132 

.140866 

.140600 

.140334 

.140068 

.139802 

.139536 

.139270 

.139005 

.138739 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 

M. 

Cosine. 

D.l”. 1 

Sine. 

D.l”. 

Cotang. 

D.l”. 

Tang. 

M. 


125 * 


54 




















































TABLE IV. LOGARITHMIC SINES, ETC, 


76 

36 * 143 * 


M. 

Sine. 

D.l". 

Cosine. 

D.l”. 

Tang. 

D.l”. 

Cotang. 

M. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 
21 
22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 
.39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 

9.769219 

.769393 

.769566 

.769740 

.769913 

.770087 

.770260 

.770433 

.770606 

.770779 

9.770952 

.771125 

.771298 

.771470 

.771643 

.771815 

.771987 

.772159 

.772331 

.772503 

9.772675 

.772847 

.773018 

.773190 

.773361 

.773533 

.773704 

.773875 

.774046 

.774217 

9.774388 

.774558 

.774729 

.774899 

.775070 

.775240 

.775410 

.775580 

.775750 

.775920 

9.776090 

.776259 

.776429 

.776598 

.776768 

.776937 

.777106 

.777275 

.777444 

.777613 

9.777781 

.777950 

.778119 

.778287 

.778455 

.778624 

.778792 

.778960 

.779128 

.779295 

.779463 

2.90 

2.90 

2.89 

2.89 

2.89 

2.89 

2.89 

2.88 

2.88 

2.88 

2.88 

2.88 

2.88 

2.87 

2.87 

2.87 

2.87 

2.87 

2.87 

2.86 

2.8C 

2.86 

2.86 

2.86 

2.85 

2.85 

2.85 

2.85 

2.85 

2.85 

2.84 

2.84 

2.84 

2.84 

2.84 

2.84 

2.83 

2.83 

2.83 

2.83 

2.83 

2.83 

2.82 

2.82 

2.82 

2.82 

2.82 

2.82 

2.81 

2.81 

2.81 

2.81 

2.81 

2.81 

2.80 

2.80 

2.80 

2.80 

2.80 

2.79 

9.907958 

.907866 

.907774 

.907682 

.907590 

.907498 

.907406 

.907314 

.907222 

.907129 

9.907037 

.906945 

.906852 

.906760 

.906667 

.906575 

.906482 

.906389 

.906296 

.906204 

9.906111 

.906018 

.905925 

.905832 

.905739 

.905645 

.905552 

.905459 

.905366 

.905272 

9.905179 

.905085 

.904992 

.904898 

.904804 

.904711 

.904617 

.904523 

.904429 

.904335 

9.904241 

.904147 

.904053 

.903959 

.903864 

.903770 

.903676 

.903581 

.903487 

.903392 

9.903298 

.903203 

.903108 

.903014 

.902919 

.902824 

.902729 

.902634 

.902539 

.902444 

.902349 

1.53 

1.53 

1.53 

1.53 

1.53 

1.53 

1.54 
1.54 
1.54 
1.54 

1.54 

1.54 

1.54 

1.54 

1.54 

1.54 

1.55 
1.55 
1.55 
1.55 

1.55 

1.55 

1.55 

1.55 

1.55 

1.55 

1.55 

1.56 
1.56 
1.56 

1.56 

1.56 

1. 5*6 
1.56 
1.56 
1.56 

1.56 

1.57 
1.57 
1.57 

1.57 

1.57 

1.57 

1.57 

1.57 

1.57 

1.57 

1.57 

1.58 
1.58 

1.58 

1.58 

1.58 

1.58 

1.58 

1.58 

1.58 

1.58 

1.59 
1.59 

9.861261 

.861527 

.861792 

.862058 

.862323 

.862589 

.862854 

.863119 

.863385 

.863650 

9.863915 

.864180 

.864445 

.864710 

.864975 

.865240 

.865505 

.865770 

.866035 

.866300 

9.866564 

.866829 

.867094 

.867358 

.867623 

.867887 

.868152 

.868416 

.868680 

.868945 

9.869209 

.869473 

.869737 

.870001 

.870265 

.870529 

.870793 

.871057 

.871321 

.871585 

9.871849 

.872112 

.872376 

.872640 

.872903 

.873167 

.873430 

.873694 

.873957 

.874220 

9.874484 

.874747 

.875010 

.875273 

.875537 

.875800 

.876063 

.876326 

.876589 

.876852 

.877114 

4.43 

4.43 

4.43 

4.42 

4.42 

4.42 

4.42 

4.42 

4.42 

4.42 

4.42 

4.42 

4.42 

4.42 

4.42 

4.41 

4.41 

4.41 

4.41 

4.41 

4.41 

4.41 

4.41 

4.41 

4.41 

4.41 

4.41 

4.41 

4.40 

4.40 

4.40 

4.40 

4.40 

4.40 

4.40 

4.40 

4.40 

4.40 

4.40 

4.40 

4.40 

4.39 

4.39 

4.39 

4.39 

4.39 

4.39 

4.39 

4.39 

4.39 

4.39 

4.39 

4.39 

4.39 

4.38 

4.38 

4.38 

4.38 

4.38 

4.38 

10.138739 

.138473 

.138208 

.137942 

.137677 

.137411 

.137146 

.136881 

.136615 

.136350 

10.136085 
.135820 
.135555 
.135290 
.135025 
.134760 
.134495 
.134230 
.133965 
.133700 

10.1 £3436 
.133171 
.132906 
.132642 
.132377 
.132113 
.131848 
.131584 
.131320 
.131055 

10.130791 

.130527 

.130263 

.129999 

.129735 

.129471 

.129207 

.128943 

.128679 

.128415 

10.128151 

.127888 

.127624 

.127360 

.127097 

.126833 

.126570 

.126306 

.126043 

.125780 

10.125516 

.125253 

.124990 

.124727 

.124463 

.124200 

.123937 

.123674 

.123411 

.123148 

.122886 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 

M. 

Cosine. 

D.l”. 

Sine. 

D.l”. 

Cotang. 

D.l”. 

Tang. 

M 


53 * 





















































37 


TABLE IV. LOGAKITHMIC SINES, ETC, 


77 


142 ° 


M. 

Sine. 

D.l" 

Cosine. 

D.l". 

Tang. 

D.l". 

Cotang. 

M. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 
. 19 

20 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 
69 
60 

9.779463 

.779631 

.779798 

.779966 

.780133 

.780300 

.780467 

.780634 

.780801 

.780968 

9.781134 

.781301 

.781468 

.781634 

.781800 

.781966 

.782132 

.782298 

.782464 

.782630 

9.782796 

.782961 

.783127 

.783292 

.783458 

.783623 

.783788 

.783953 

.784118 

.784282 

9.784447 

.784612 

.784776 

.784941 

.785105 

.785269 

.785433 

.785597 

.785761 

.785925 

9.786089 

.786252 

.786416 

.786579 

.786742 

.786906 

.787069 

.787232 

.787395 

.787557 

9.787720 

.787883 

.788045 

.788208 

.788370 

.788532 

.788694 

.788856 

.789018 

.789180 

.789342 

2.79 

2.79 

2.79 

2.79 

2.79 

2.78 

2.78 

2.78 

2.78 

2.78 

2.78 

2.77 

2.77 

2.77 

2.77 

2.77 

2.77 

2.76 

2.76 

2.76 

2.76 

2.76 

2.76 

2.75 

2.75 

2.75 

2.75 

2.75 

2.75 

2.74 

2.74 

2.74 

2.74 

2.74 

2.74 

2.73 

2.73 

2.73 

2.73 

2.73 

2.73 

2.73 

2.72 

2.72 

2.72 

2.72 

2.72 

2.72 

2.71 

2.71 

2.71 

2.71 

2.71 

2.71 

2.70 

2.70 

2.70 

2.70 

2.70 

2.70 

9.902349 

.902253 

.902158 

.902063 

.901967 

.901872 

.901776 

.1W11681 

.901585 

.901490 

9.901394 

.901298 

.901202 

.901106 

.901010 

.900914 

.900818 

.900722 

.900626 

.900529 

9.900433 

.900337 

.900240 

.900144 

.900047 

.899951 

.899854 

.899757 

.899660 

.899564 

9.899467 

.899370 

.899273 

.899176 

.899078 

.898981 

.898884 

.898787 

.898689 

.898592 

9.898494 

.898397 

.898299 

.898202 

.898104 

.898006 

.897908 

.897810 

.897712 

.897614 

9.897516 

.897418 

.897320 

.897222 

.897123 

.897025 

.896926 

.896828 

.896729 

.896631 

.896532 

1.59 

1.59 

1.59 

1.59 

1.59 

1.59 

1.59 

1.59 

1.59 

1.60 

1.60 

1.60 

1.60 

1.60 

1.60 

1.60 

1.60 

1.60 

1.60 

1.61 

1.61 

1.61 

1.61 

1.61 

1.61 

1.61 

1.61 

1.61 

1,61 

1.62 

1.62 

1.62 

1.62 

1.62 

1.62 

1.62 

1.62 

1.62 

1.62 

1.62 

1.63 

1.63 

1.63 

1.63 

1.63 

1.63 

1.63 

1.63 

1.63 

1.63 

1.64 
1.64 
1.64 
1.64 
1.64 
1.64 
1.64 
1.64 
1.64 
1.64 

9.877114 

.877377 

.877640 

.877903 

.878165 

.878428 

.878691 

.878953 

.879216 

.879478 

9.879741 

.880003 

.880265 

.880528 

.880790 

.881052 

.881314 

.881577 

.881839 

.882101 

9.882363 

.882625 

.882887 

.883148 

.883410 

.883672 

.883934 

.884196 

.884457 

.884719 

9.884980 

.885242 

.885504 

.885765 

.886026 

.886288 

.886549 

.886811 

.887072 

.887333 

9.887594 

.887855 

.888116 

.888378 

.888639 

.888900 

.889161 

.889421 

.889682 

.889943 

9.890204 
.890465 
.890725 
.890986 
.891247 
.891507 
.891768 
.892028 
.892289 
.892549 
.892810 

4.38 

4.38 

4.38 

4.38 

4.38 

4.38 

4.38 

4.38 

4.37 

4.37 

4.37 

4.37 

4.37 

4.37 

4.37 

4.37 

4.37 

4.37 

4.37 

4.37 

4.37 

4.37 

4.36 

4.36 

4.36 

4.36 

4.36 

4.36 

4.36 

4.36 

4.36 

4.36 

4.36 

4.36 

4.36 

4.36 

4.36 

4.35 

4.35 

4.35 

4.35 

4.35 

4.35 

4.35 

4.35 

4.35 

4.35 

4.35 

4.35 

4.35 

4.35 

4.35 

4.34 

4.34 

4.34 

4.34 

4.34 

4.34 

4.34 

4.34 

10.122886 

.122623 

.122360 

.122097 

.121835 

.121572 

.121309 

.121047 

.120784 

.120522 

10.120259 

.119997 

.119735 

.119472 

.119210 

.118948 

.118686 

.118423 

.118161 

.117899 

10.117637 

.117375 

.117113 

.116852 

.116590 

.116328 

.116066 

.115804 

.115543 

.115281 

10.115020 

.114758 

.114496 

.114235 

.113974 

.113712 

.113451 

.113189 

.112928 

.112667 

10.112406 

.112145 

.111884 

.111622 

.111361 

.111100 

.110839 

.110579 

.110318 

.110057 

10.109796 
.109535 
.109275 
.101X114 
.108753 
.108493 
.108232 
.107972 
.107711 
.107451 
.107190 

60 

69 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 

M. 

Cosine. 

D.l". 

Sine. 

D.l". 

Cotaiig. 

D.l". 

Tang. 

M. 


127 * 


52 
















































78 TABLE IV. LOGARITHMIC SINES, ETC. 


141 


as 


M. 

Sine. 

D.l". 

Cosine. 

D.l". 

Tang. 

D.l". 

Cotang. 

M. | 

0 

1 

2 

3 

4 

5 

G 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 
21 
22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 
64 

55 

56 

57 

58 

59 

60 

9.789342 

.789504 

.789665 

.789827 

.789988 

.790149 

.790310 

.790471 

.790632 

.790793 

9.790954 

.791115 

.791275 

.791436 

.791596 

.791757 

.791917 

.792077 

.792237 

.792397 

9.792557 

.792716 

.792876 

.793035 

.793195 

.793354 

.793514 

.793673 

.793832 

.793991 

9.794150 

.794308 

.794467 

.794626 

.794784 

.794942 

.795101 

.795259 

.795417 

.795575 

9.795733 

.795891 

.796049 

.796206 

.796364 

.796521 

.796679 

.796836 

.796993 

.797150 

9.797307 

.797464 

.797621 

.797777 

.797934 

.798091 

.798247 

.798403 

.798560 

.798716 

.798872 

2.69 

2.69 

2.69 

2.69 

2.69 

2.69 

2 68 
2.68 
2.68 
2.68 

2.68 

2.68 

2.67 

2.67 

2.67 

2.67 

2.67 

2.67 

2.67 

2.66 

2.66 

2.66 

2.66 

2.66 

2.66 

2.65 

2.65 

2.65 

2.65 

2.65 

2.65 

2.64 

2.64 

2.64 

2.64 

2.64 

2.64 

2.64 

2.63 

2.63 

2.63 

2.63 

2.63 

2.63 

2.62 

2.62 

2.62 

2.62 

2.62 

2.61 

2.61 

2.61 

2.61 

2.61 

2.61 

2.61 

2.61 

2.60 

2.60 

2.60 

9.896532 

.896433 

.896335 

.896236 

.896137 

.896038 

.895939 

.895840 

.895741 

.895641 

9.895542 

.895443 

.895343 

.895244 

.895145 

.895045 

.894945 

.894846 

.894746 

.894646 

9.894546 

.894446 

.894346 

.894246 

.894146 

.894046 

.893946 

.893846 

.893745 

.893645 

9.893544 

.893444 

.893343 

.893243 

.893142 

.893041 

.892940 

.892839 

.892739 

.892638 

9 892536 
.892435 
.892334 
.892233 
.892132 
.892030 
.891929 
.891827 
.891726 
.891624 

9.891523 

.891421 

.891319 

.891217 

.891115 

.891013 

.890911 

.890809 

.890707 

.890605 

.890503 

1.65 

1.65 

1.65 

1.65 

1.65 

1.65 

1.65 

1.65 

1.65 

1.65 

1.66 
1.66 
1.66 
1.66 
1.66 

1.66 
1.66 
1.66 

1.66 
1.66 

1.67 

1.67 

1.67 

1.67 

1.67 

1.67 

1.67 

1.67 

1.67 

1.67 

1.68 
1.68 

1.68 
1.68 
1.68 
1.68 

1.68 
1.68 
1.68 
1.68 

1.69 

1.69 

1.69 

1.69 

1.69 

1.69 

1.69 

1 69 

1 69 

1.69 

1.70 
1.70 
1.70 
1.70 
1.70 
1.70 
1.70 
1.70 
1.70 
1.70 

9.892810 

.893070 

.893331 

.893591 

.893851 

.894111 

.894372 

.894632 

.894892 

.895152 

9.895412 

.895672 

.895932 

.896192 

.896452 

.896712 

.896971 

.897231 

.897491 

.897751 

9.898010 

.898270 

.898530 

.898789 

.899049 

.899308 

.899568 

.899827 

.900086 

.900346 

9.900605 

.900864 

.901124 

.901383 

.901642 

.901901 

.902100 

.902420 

.902679 

.902938 

9.903197 

.903456 

.903714 

.903973 

.904232 

.904491 

.904750 

.905008 

.905267 

.905526 

9.905785 

.906043 

.906302 

.906560 

.906819 

.907077 

.907336 

.907594 

.907852 

.908111 

.908369 

4.34 

4.34 

4.34 

4.34 

4.34 

4.34 

4.34 

4.34 

4.33 

4.33 

4.33 

4.33 

4.33 

4.33 

4.33 

4.33 

4.33 

4.33 

4.33 

4.33 

4.33 

4.33 

4 33 
4.33 
4.33 
4.32 
4.32 
4.32 
4.32 
4.32 

4.32 

4.32 

4.32 

4.32 

4.32 

4.32 

4.32 

4.32 

4.32 

4.32 

4.32 

4.32 

4.31 

4.31 

4.31 

4.31 

4.31 

4.31 

4.31 

4.31 

4.31 

4.31 

4.31 

4.31 

4.31 

4.31 

4.31 

4.31 

4.31 

4.31 

10.107190 
.106930 
.106669 
.106409 
.106149 
.105889 
.105628 
.105368 
.105108 
.104848 

10.104588 
.104329 
.104068 
.103808 
.103548 
.103288 
.103029 
.102769 
.102509 
.102249 

10.101990 
.101730 
.101470 
.101211 
.100951 
.100692 
.100432 
.100173 
.099914 
.099654 

10.099395 

.099136 

.098876 

.098617 

.098358 

.098099 

.097840 

.097580 

.097321 

.097062 

10.096803 

.096544 

.096286 

.096027 

.095768 

.095509 

.095250 

.094992 

.094733 

.094474 

10.094215 

.093957 

.093698 

.093440 

.093181 

.092923 

.092664 

.092406 

.092148 

.091889 

.091631 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 

M. 

Cosine. 

D.l". 

Sine. 

D.l" 

. Cotang. 

D.l". 

Tang. 

M. 
















































































TABLE IV. LOGARITHMIC SINES, ETC 


79 

39 * 140 


M. 

Sine. 

P.l ". 

Cosine. 

D. 1 '. 

Tang. 

P.l''. 

Cotang. 

M 

0 

1 

2 

3 

4 

5 

G 

7 

8 
9 

10 

11 

12 

13 

14 

15 
1G 

17 

18 

19 

20 
21 
22 

23 

24 

25 
2G 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 
4G 

47 

48 

49 

50 

51 

52 

53 
64 

55 

56 

57 

58 

59 
GO 

9.798872 

.799028 

.799184 

.799339 

.799495 

.799651 

.799806 

.799962 

.800117 

.800272 

9.800427 

.800582 

.800737 

.800892 

.801047 

.801201 

.801356 

.801511 

.801665 

.801819 

9.801973 

.802128 

.802282 

.802436 

.802589 

.802743 

.802897 

.803050 

.803204 

.803357 

9.803511 

.803664 

.803817 

.803970 

.804123 

.804276 

.804428 

.804581 

.804734 

.804886 

9.805039 

.805191 

.805343 

.805495 

.80.5647 

.805799 

.805951 

.806103 

.806254 

.806406 

9.806557 

.806709 

.806860 

.807011 

.807163 

.807314 

.807465 

.807615 

.807766 

.807917 

.808067 

2.60 

2.60 

2.60 

2.59 

2.59 

2.59 

2.59 

2.59 

2.59 

2.59 

2.58 

2.58 

2.58 

2.58 

2.58 

2.58 

2.57 

2.57 

2.57 

2.57 

2.57 

2.57 

2.57 

2.56 

2.56 

2.56 

2.56 

2.56 

2.56 

2.55 

2.55 

2.55 

2.55 

2.55 

2.55 

2.55 

2.54 

2.54 

2.54 

2.54 

2.54 

2 54 
2.54 
2.53 
2.53 
2.53 
2.53 
2.53 
2.53 
2.52 

2.52 

2.52 

2.52 

2.52 

2.52 

2.52 

2.51 

2.51 

2.51 

2.51 

9.890503 

.890400 

.890298 

.890195 

.890093 

.889990 

.889888 

.889785 

.889682 

.889579 

9.889477 

.889374 

.889271 

.889168 

.889064 

.888961 

.888858 

.888755 

.888651 

.888548 

9.888444 

.888341 

.888237 

.888134 

.888030 

.887926 

.887822 

.887718 

.887614 

.887510 

9.887406 

.887302 

.887198 

.887093 

.886989 

.886885 

.886780 

.886676 

.886571 

.886466 

9.886362 

.886257 

.886152 

.886047 

.885942 

.885837 

.885732 

.885627 

.885522 

.885416 

9.885311 

.885205 

.885100 

.884994 

.884889 

.884783 

.884677 

.884572 

.884466 

.884360 

.884254 

1.71 

1.71 

1.71 

1.71 

1.71 

1.71 

1.71 

1.71 

1.71 

1.71 

1.72 
1.72 

1 72 
1.72 
1.72 
1.72 
1.72 
1.72 
1.72 

1.72 

1.73 
1.73 
1.73 
1.73 
1.73 
1.73 
1.73 
1.73 

1.73 

1.74 

1.74 

1.74 

1.74 

1.74 

1.74 

1.74 

1.74 

1.74 

1 74 

1.75 

1.75 

1.75 
1.75 
1.75 
1.75 
1.75 
1.75 
1.75 

1.75 

1.76 

1.76 

1.76 

1 76 
1.76 
1.76 
1.76 
1.76 

1.76 

1.77 
1.77 

9.908369 

.908628 

.908886 

.909144 

.909402 

.909660 

.909918 

.910177 

.910435 

.910693 

9.910951 

.911209 

.911467 

.911724 

.911982 

.912240 

.912498 

.912756 

.913014 

.913271 

9.913529 

.913787 

.914044 

.914302 

.914560 

.914817 

.915075 

.915332 

.915590 

.915847 

9.916104 

.916362 

.916619 

.916877 

.917134 

.917391 

.917648 

.917906 

.918162 

.918420 

9.918677 

.918934 

.919191 

.919448 

.919705 

.919962 

.920219 

.920476 

.920733 

.920990 

9.921247 

.921503 

.921760 

.922017 

.922274 

.922530 

.922787 

.923044 

.923300 

.923557 

.923813 

4.30 

4.30 

4.30 

4.30 

4.30 

4.30 

4.30 

4.30 

4.30 

4.30 

4.30 

4.30 

4.30 

4.30 

4.30 

4.30 

4.30 

4.30 

4.30 

4.30 

4.29 

4.29 

4.29 

4.29 

4.29 

4.29 

4.29 

4.29 

4.29 

4.29 

4.29 

4.29 

4.29 

4.29 

4.29 

4.29 

4.29 

4.29 

4.29 

4.29 

4.28 

4.28 

4.28 

4.28 

4.28 

4.28 

4.28 

4.28 

4.28 

4.28 

4.28 

4.28 ■ 

4.28 

4.28 

4.28 

4.28 

4.28 

4.28 

4.28 

4.28 

10.091631 

.091372 

.091114 

.090856 

.090598 

.090340 

.090082 

.089823 

.08956'j 

.089307 

10.089049 

.088791 

.088533 

.088276 

.088018 

.087760 

.087502 

.087244 

.086986 

.086729 

10.086471 

.086213 

085953 

.085698 

085440 

.085183 

.084925 

.084668 

.084410 

.084153 

10.083896 

.083638 

.083381 

.083123 

.082866 

.082609 

.082352 

.082094 

.081838 

.081580 

10.081323 

.081066 

.080809 

.080552 

.080295 

.080038 

.079781 

.079524 

.079267 

.079010 

10.078753 

.078497 

.078240 

.077983 

.077726 

.077470 

.077213 

.076956 

.076700 

.076443 

.076187 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

o 

1 

0 

M. 

Cosine. 

D. 1". 

Sine. 

D. 1". 

Cotang. 

D. 1". 

Tang. 

M. 





































TABLE IV. LOGARITHMIC SINES, ETC, 


13ft 


80 


40° 


M. 

Sine. 

D.l". 

Cosine. 

D.l". 

Tang. 

D.l". 

Cotang. 

M. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 
21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 

9.808067 
.808218 
.808368 
.808519 
.808669 
.808819 
.808969 
809119 
.809269 
.809419 

9.809569 

.809718 

.809868 

.810017 

.810167 

.810316 

.810465 

.810614 

.810763 

.810912 

9.811061 

.811210 

.811358 

.811507 

.811655 

.811804 

.811952 

.812100 

.812248 

.812396 

9.812544 

.812692 

.812840 

.812988 

.813135 

.813283 

.813430 

.813578 

.813725 

.813872 

9.814019 

.814166 

.814313 

.814460 

.814607 

.814753 

.814900 

.815046 

.815193 

.815339 

9.815485 

.815631 

.815778 

.815924 

.816069 

.816215 

.816361 

.816507 

.816652 

.816798 

.816943 

2.51 

2.51 

2.51 

2.50 

2.50 

2.50 

2.50 

2.50 

2.50 

2.50 

2.49 

2.49 

2.49 

2.49 

2.49 

2.49 

2.48 

2.48 

2.48 

2.48 

2.48 

2.48 

2.48 

2.47 

2.47 

2.47 

2.47 

2.47 

2.47 

2.47 

2.46 

2.46 

2.46 

2.46 

2.46 

2.46 

2.46 

2.45 

2.45 

2.45 

2.45 

2.45 

2.45 

2.45 

2.44 

2.44 

2.44 

2.44 

2.44 

2.44 

2.44 

2.43 

2.43 

2.43 

2.43 

2.43 

2.43 

2.43 

2.42 

2.42 

9.884254 

.884148 

.884042 

.883936 

.883829 

.883723 

.883617 

.883510 

.883404 

.883297 

9.883191 

.88:1084 

.882977 

.882871 

.882764 

.882657 

.882550 

.882443 

.882336 

.882229 

9.882121 

.882014 

.881907 

.881799 

.881692 

.881584 

.881477 

.881369 

.881261 

.881153 

9.881046 

.880938 

.880830 

.880722 

.880613 

.880505 

.880397 

.880289 

.880180 

.880072 

9.879963 

.879855 

.879746 

.879637 

.879529 

.879420 

.879311 

.879202 

.879093 

.878984 

9.878875 

.878766 

.878656 

.878547 

.878438 

.878328 

.878219 

.878109 

.877999 

.877890 

.877780 

1 77 
1.77 
1.77 
1.77 
1.77 
1.77 
1.77 

1.77 

1.78 
1.78 

1.78 

1.78 

1.78 

1.78 

1.78 

1.78 

1.78 

1.79 
1.79 
1.79 

1.79 

1.79 

1.79 

1.79 

1.79 

1.79 

1.79 

1.80 
1.80 
1.80 

1.80 

1.80 
1.80 
1.80 
1.80 
1.80 

1.81 
1.81 
1.81 
1.81 

1.81 

1.81 

1.81 

1.81 
1.81 
1.81 
1.82 
1.82 
1.82 

1.82 

1.82 

1.82 

1.82 

1.82 

1 .82 
1.83 
1.83 
1.83 
1.83 
1.83 

9.923813 

.924070 

.924327 

.924583 

.924840 

.925096 

.925352 

.925609 

.925865 

.926122 

9.926378 

.926634 

.926890 

.927147 

.927403 

.927659 

.927915 

.928171 

.928427 

.928683 

9.928940 

.929196 

.929452 

.929708 

.929964 

.930220 

.9:10475 

.930731 

.930987 

.931243 

9.931499 

.931755 

.932010 

.932266 

.932522 

.932778 

.933033 

.933289 

.933545 

.9:53800 

9.934056 

.934311 

.934567 

.934823 

.935078 

.935333 

.935589 

.935844 

.936100 

.936355 

9.936611 

.936866 

.937121 

.937376 

.937632 

.937887 

.938142 

.938398 

.938653 

.938908 

.939163 

4.28 

4.28 

4 27 
4.27 
4.27 
4.27 
4.27 
4.27 
4.27 
4.27 

4.27 

4.27 

4.27 

4.27 

4.27 

4.27 

4.27 

4.27 

4.27 

4.27 

4.27 

4.27 

4.27 

4.27 

4.27 

4.27 

4.26 

4.26 

4.26 

4.26 

4.26 

4.26 

4.26 

4.26 

4.26 

4.26 

4.26 

4.26 

4 26 
4.26 

4.26 

4.26 

4.26 

4.26 

4.26 

4.26 

4.26 

4.26 

4.26 

4.26 

4 26 
4.26 
4.26 
4.25 
4.25 
4.25 
4.25 
4.25 
4.25 
4.25 

10.076187 

.075930 

.075673 

.075417 

.075160 

.074904 

.074648 

.074391 

.074135 

.073878 

10.073622 

.073366 

.073110 

.072853 

.072597 

.072341 

.072085 

.071829 

.071573 

.071317 

10.071060 

.070804 

.070548 

.070292 

.070036 

.069780 

.069525 

.069269 

.069013 

.068757 

10.068501 

.068245 

.067990 

.067734 

.067478 

.067222 

.066967 

.066711 

.066455 

.066200 

10.065944 

.065689 

.065433 

.065177 

.064922 

.064667 

.064411 

.064156 

.063900 

.063645 

10.063389 

.063134 

.062879 

.062624 

.062368 

.062113 

.061858 

.061602 

.061347 

.061092 

.060837 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

l 

0 

M. 

Cosine. 

D.l". 

Sine. 

D.l". 

Cotang. 

D.l". 

Tang. 

M. 


130° 49° 

































41 


TABLE IV. LOGARITHMIC SINES, ETC, 


81 


138* 


M. 

Sine. 

D.i . 

Cosine. 

D.I'. 

Tang. 

D.I". 

Cotang. 

M. 

0 

1 

2 

3 

4 

5 

G 

7 

8 

9 

10 

It 

12 

13 

14 

15 

16 

17 

18 

19 

20 
21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 

9.816943 

.817088 

.817233 

.817379 

.817524 

.817668 

.817813 

.817958 

.818103 

.818247 

9.818392 

.818536 

.818681 

.818825 

.818969 

.819113 

.819257 

.819401 

.819545 

.819689 

9.819832 

.819976 

.820120 

.820263 

.820406 

.820550 

.820693 

.820836 

.820979 

.821122 

9.821265 

.821407 

.821550 

.821693 

.821835 

.821977 

.822120 

.822262 

.822404 

.822546 

9.822688 

.822830 

.822972 

.823114 

.823255 

.823397 

.823539 

.823680 

.823821 

.823963 

9.824104 

.824245 

.824386 

.824527 

.824668 

.824808 

.824949 

.825090 

.825230 

.825371 

.825511 

2.42 

2.42 

2.42 

2.42 

2.42 

2.41 

2.41 

2.41 

2.41 

2.41 

2.41 

2.41 

2.40 

2.40 

2.40 

2.40 

2.40 

2.40 

2.40 

2.39 

2.39 

2.39 

2.39 

2.39 

2.39 

2.39 

2.38 

2.38 

2.38 

2.38 

2.38 

2.38 

2.38 

2 37 
2.37 
2.37 
2.37 
2.37 
2.37 
2.37 

2.37 

2.36 

2.36 

2.36 

2.36 

2.36 

2.36 

2.36 

2.35. 

2.35 

2.35 

2.35 

2.35 

2.35 

2.35 

2.34 

2.34 

2.34 

2.34 

2.34 

9.877780 

.877670 

.877560 

.8774.50 

.877340 

.877230 

.877120 

.877010 

.876899 

.876789 

9.876678 

.876568 

.876457 

.876347 

.876236 

.876125 

.876014 

.875904 

.875793 

.875682 

9.875571 

.875459 

.875348 

.875237 

.875126 

.875014 

.874903 

.874791 

.874680 

.874568 

9.874456 

.874344 

.874232 

.874121 

.874009 

.873896 

.873784 

.873672 

.873560 

.873448 

9.873335 

.873223 

.873110 

.872998 

.872885 

.872772 

.872659 

.872547 

.872434 

.872321 

9.872208 

.872095 

.871981 

.871868 

.871755 

.871641 

.871528 

.871414 

.871301 

.871187 

.871073 

1 83 
1.83 
1.83 

1.83 

1.84 
1.84 
1.84 
1.84 
1.84 
1.84 

1.84 

1.84 

1.84 

1.84 

1.85 
1.85 
1.85 
1.85 
1.85 
1.85 

1.85 

1.85 

1.85 

1.86 
1.86 
1.86 
1.86 

1.86 

1.86 
1.86 

1.86 

1 .86 
1.87 
1.87 
1.87 
1.87 
1.87 
1.87 
1.87 
1.87 

1.87 

1.88 
1.88 

1.88 

1.88 

1.88 

1.88 

1.88 

1.88 

1.88 

1.89 

1.89 

1.89 

1.89 

1.89 

1.89 

1.89 

1.89 

1.89 

1.90 

9.939163 

.939418 

.939(573 

.939928 

.940183 

.940438 

.940694 

.940949 

.941204 

.941458 

9.941713 

.941968 

.942223 

.942478 

.942733 

.942988 

.943243 

.943498 

.943752 

.944007 

9.944262 

.944517 

.944771 

.945026 

.945281 

.945535 

.945790 

.946045 

.946299 

.946554 

9.946808 

.947063 

.947318 

.947572 

.947826 

.948081 

.948336 

.948590 

.948844 

.949099 

9.949353 

.949608 

.949862 

.950116 

.950371 

.950625 

950879 

.951133 

.951388 

.951642 

9.951896 

.952150 

.952405 

.952659 

.952913 

.953167 

.953421 

.953675 

.953929 

.954183 

.954437 

4.25 

4.25 

4.25 

4.25 

4.25 

4.25 

4.25 

4.25 

4.25 

4.25 

4.25 

4.25 

4.25 

4.25 

4.25 

4.25 

4.25 

4.25 

4.25 

4.25 

4.25 

4.25 

4.24 

4.24 

4.24 

4.24 

4.24 

4.24 

4.24 

4.24 

4.24 

4.24 

4.24 

4.24 

4.24 

4.24 

4.24 

4.24 

4 24 
4.24 

4.24 

4.24 

4.24 

4.24 

4.24 

4.24 

4.24 

4.24 

4.24 

4.24 

4 24 
4.24 
4.24 
4.24 
4.24 
4.24 
4.24 
4.23 
4.23 
4.23 

10.060837 

.060582 

.060327 

.060072 

.059817 

.059562 

.059306 

.059051 

.058796 

.058542 

10.058287 

.058032 

.057777 

.057522 

.057267 

.057012 

.056757 

.056502 

.056248 

.055993 

10.055738 

.055483 

.055229 

.054974 

.054719 

.054465 

.054210 

.053955 

.053701 

.053446 

10.053192 

.052937 

.052682 

.052428 

.052174 

.051919 

.051664 

.051410 

.051156 

.050901 

10.050647 

.050392 

.050138 

.049884 

.049629 

.049375 

.049121 

.048867 

.048612 

.048358 

10.048104 
.047850 
.047595 
.04734t 
.047087 
.046833 
.046579 
.046325 
.046071 
.045817 
.045563 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 

M. 

Cosine. 

D,l". 

Sine. 

D.I”. 

Cotang. 

D.I". 

Tang. 

M. 


131° 48* 










































TABLE IV. LOGAltITlIMIC SINES, ETC. 


137 


82 


42” 


M. 

Sine. 

D. 1 ”. 

Cosine. 

D.l”. 

Tang. 

D.l”. 

Cotang. 

M. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 
21 
22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 

9.825511 

.825651 

.825791 

.825931 

.826071 

.826211 

.826351 

.826491 

.826631 

.826770 

9.826910 

.827049 

.827189 

.827328 

.827467 

.827606 

.827745 

.827884 

.828023 

.828162 

9.828301 

.828439 

.828578 

.828716 

.828855 

.828993 

.829131 

.829269 

.829407 

.829545 

9.829683 

.829821 

.829959 

.830097 

.830234 

.830372 

.830509 

.830646 

.830784 

.830921 

9.831058 

.831195 

.831332 

.831469 

.831606 

.831742 

.831879 

.832015 

.832152 

.832288 

9.832425 

.832561 

.832697 

.832833 

.832969 

.833105 

.833241 

.833377 

.833512 

.833648 

.833783 

2.34 

2.34 

2.33 

2.33 

2.33 

2.33 

2.33 

2.33 

2.33 

2.33 

2.32 

2.32 

2.32 

2.32 

2.32 

2.32 

2.32 

2.31 

2.31 

2.31 

2.31 

2.31 

2.31 

2.31 

2.31 

2.30 

2.30 

2.30 

2.30 

2.30 

2.30 
2.30 
2.29 
2.29 
. 2.29 
2.29 
2.29 
2.29 
2.29 
2.29 

2.28 

2.28 

2.28 

2.28 

2.28 

2.28 

2.28 

2.27 

2.27 

2.27 

2.27 

2.27 

2.27 

2.27 

2.27 

2.26 

2.26 

2.26 

2.26 

2.26 

9.871073 

.870960 

.870846 

.870732 

.870618 

.870504 

.870390 

.870276 

.870161 

.870047 

9.869933 

.869818 

.869701 

.869589 

.869474 

.869360 

.869245 

.869130 

.869015 

.868900 

9.868785 

.868670 

.868555 

.868440 

.868324 

.868209 

.SOXOiCi 

.867978 

.867862 

.867747 

9.867631 

.867515 

.867399 

.867283 

.867167 

.867051 

.866935 

.866819 

.866703 

.866586 

9.866470 

.866353 

.866237 

.866120 

.866004 

.865887 

.865770 

.865653 

.865536 

.865419 

9.865302 
.865185 
.865068 
.864950 
.864833 
.864716 
.864598 
.864481 
.864363 
.864245 
.864127 

1.90 

1.90 

1.90 

1.90 

1.90 

1.90 

1.90 

1.90 

1.91 
1.91 

1.91 

1.91 

1.91 

1.91 

1.91 

1.91 

1.91 

1.92 
1.92 
1.92 

1.92 

1.92 

1.92 

1.92 

1.92 

1.92 

1.93 
1.93 
1.93 

1 93 

1.93 

1.93 

1.93 

1.93 

1.93 

1.94 
1.94 
1.94 
1.94 
1.94 

1.94 

1.94 

1.94 

1.94 

1.95 
1.95 
1.95 
1.95 
1.95 
1.95 

1.95 

1.95 

1.95 

1.96 
1.96 
1.96 
1.96 
1.96 
1.96 
1.96 

9.954437 

.954691 

.954945 

.955200 

.955454 

.955707 

.955961 

.956215 

.956469 

.956723 

9.956977 

.957231 

.957485 

.957739 

.957993 

.958246 

.958500 

.958754 

.959008 

.959262 

9.959516 

.959769 

.960023 

.960277 

.960530 

.960784 

.961038 

.961292 

.961545 

.961799 

9.962052 

.962306 

.962560 

.962813 

.963067 

.963320 

.963574 

.963828 

.964081 

.964335 

9.964588 

.964842 

.965095 

.965349 

.965602 

.965855 

.966109 

.966362 

.966616 

.966869 

9.967123 

.967376 

.967629 

.967883 

.968136 

.968389 

.968643 

.968896 

.969149 

.969403 

.969656 

4.23 

4.23 

4.23 

4.23 

4.23 

4.23 

4.23 

4.23 

4.23 

4.23 

4.23 

4.23 

4.23 

4.23 

4.23 

4.23 

4.23' 

4.23 

4 23 
4.23 

4.23 

4.23 

4.23 

4.23 

4.23 

4.23 

4.23 

4.23 

4.23 

4.23 

4.23 

4.23 

4.23 

4.23 

4.23 

4.23 

4.23 

4.23 

4.23 

4.23 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

10.045563 

.045309 

.045055 

.044800 

.044546 

.044293 

.044039 

.043785 

.043531 

043277 

10.043023 

.042769 

.042515 

.042261 

.042007 

.041754 

.041500 

.041246 

.040992 

.040738 

10.010484 

.040231 

.039977 

.039723 

.039470 

.039216 

.038962 

.038708 

.038455 

.038201 

10.037948 

.037694 

.037440 

.037187 

.036933 

.036680 

.036426 

.036172 

.035919 

.035665 

10.035412 

.035158 

.034905 

.034651 

.034398 

.034145 

.033891 

.033638 

.033384 

.033131 

10.032877 

.032624 

.032371 

.032117 

.031864 

.031611 

.031357 

.031104 

.030851 

.030597 

.030344 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 

M. 

Cosine, 

D.l 

Sine. 

D.l”. 

Cotang. 

D.l”. 

Tang 

M. 


132° . . 47» 



































TABLE IV. LOGARITHMIC SINES, ETC 


83 


1 » 6 ° 


M. 

Sine. 

D. 1”. 

Cosine. 

D.l 

Tang. 

D. l". 

Cotang. 

M. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 
21 
22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 

9.833783 

.833919 

.834054 

.834189 

.834325 

.834460 

.834595 

.834730 

•834865 

.834999 

9.835134 

.835269 

.835403 

.835538 

.835672 

.835807 

.835941 

.836075 

.836209 

.836343 

9.836477 

.836611 

.836745 

.836878 

.837012 

.837146 

.837279 

.837412 

.837546 

.837679 

9.837812 

.837945 

.838078 

.838211 

.838344 

.838477 

.838610 

.838742 

.838875 

.839007 

9.839140 

.839272 

.839401 

.839536 

.839668 

.839800 

.839932 

.840064 

.840196 

.840328 

9.840459 

.840591 

.840722 

.840854 

.840985 

.841116 

.841247 

.841378 

.841509 

.841640 

.841771 

2.26 

2.26 

2.25 

2.25 

2.25 

2.25 

2.25 

2.25 

2.25 

2.25 

2.24 

2.24 

2.24 

2.24 

2.24 

2.24 

2.24 

2.23 

2.23 

2.23 

2.23 

2.23 

2.23 

2.23 

2.23 

2.22 

2.22 

2.22 

2.22 

2.22 

2.22 

2.22 

2.22 

2.21 

2.21 

2.21 

2.21 

2.21 

2.21 

2.21 

2.21 

2.20 

2.20 

2.20 

2.20 

2.20 

2.20 

2.20 

2.19 

2.19 

2.19 

2.19 

2.19 

2.19 

2.19 

2.19 

2.18 

2.18 

2.18 

2.18 

9.864127 

.864010 

.863892 

.863774 

.863656 

.863538 

.863419 

.863301 

.863183 

.863064 

9.862946 

.862827 

.862709 

.862590 

.862471 

.862353 

.862234 

.862115 

.861996 

.861877 

9.861758 

.861638 

.861519 

.861400 

.861280 

.861161 

.861041 

.860922 

.860802 

.860682 

9.860562 

.860442 

.860322 

.860202 

.860082 

.859962 

.859842 

.859721 

.859601 

.859480 

9 859300 
.859239 
.859119 
.858998 
.858877 
.858756 
.858635 
.858514 
.858393 
.858272 

9.858151 

.858029 

.857908 

.857786 

.857665 

.857543 

.857422 

.857300 

.857178 

.857056 

.856934 

1.96 

1.97 
1.97 
1.97 
1.97 
1.97 
1.97 
1.97 
1.97 

1.97 

1.98 
1.98 
1.98 
1.98 
1.98 
1.98 
1.98 
1.98 

1.98 

1.99 

1.99 

1.99 

1.99 

1.99 

1.99 

1.99 

1.99 

2.00 

2.00 

2.00 

2.00 

2.00 

2.00 

2.00 

2.00 

2.00 

2.01 

2.01 

2.01 

2.01 

2.01 

2.01 

2.01 

2.01 

2.02 

2.02 

2.02 

2.02 

2.02 

2.02 

2.02 

2.02 

2.02 

2.03 

2.03 

2.03 

2.03 

2.03 

2.03 

2.03 

9.969656 

.969909 

.970162 

.970416 

.970669 

.970922 

.971175 

.971429 

.971682 

.971935 

9.972188 

.972441 

.972694 

.972948 

.973201 

.973454 

.973707 

.973960 

.974213 

974466 

9.974720 

.974973 

.975226 

.975479 

.975732 

.975985 

.976238 

.976491 

.976744 

.976997 

9.977250 

.977503 

.977756 

.978009 

.978262 

.978515 

.978768 

.979021 

.979274 

.979527 

9.979780 

.980033 

.980286 

.980538 

.980791 

.981044 

.981297 

.981550 

.981803 

.982056 

9.982309 

.982562 

.982814 

.983067 

.983320 

.983573 

.983826 

984079 

.984331 

.984584 

.984837 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.21 

4.21 

4.21 

4 21 
4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

10.030344 

.030091 

.029838 

.029584 

.029331 

.029078 

.028825 

.028571 

.028318 

.028065 

10.027812 

.027559 

.027306 

.027052 

.026799 

.026546 

.026293 

.020040 

.025787 

.025534 

10.025280 

.025027 

.024774 

.024521 

.024268 

.024015 

.023762 

.023509 

.023256 

.023003 

10.022750 

.022497 

.022244 

":021991 

.021738 

.021485 

.021232 

.020979 

.020726 

.020473 

10.020220 

.019967 

.019714 

.019462 

.019209 

.018956 

.018703 

.018450 

.018197 

.017944 

10.017691 

.017438 

.017186 

.016933 

.016680 

.016427 

.016174 

.015921 

.015669 

.015416 

.015163 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 

M. 

Cosine. 

D. 1 

Sine. 

D.l". 

Cotang. 

D. 1". 

Tang. 

M. 





































table iv. logarithmic sines, etc. 


135 


84 


44° 


M. 

Sine. 

D. 1”. 

Cosine. 

I). 1". 

Tang. 

D. 1 

Cotang. 

M. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 
10 

17 

18 

19 

20 
21 
22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 

9.841771 

.841902 

.842033 

.842163 

.842294 

.842424 

.842555 

.842685 

.842815 

.842946 

9.843070 

.843206 

.843336 

.843466 

.843595 

.843725 

.843855 

.843984 

.844114 

.844243 

9.844372 

.844502 

.844631 

.844760 

.844889 

.845018 

.845147 

.845276 

.845405 

.845533 

9.845662 

.845790 

.845919 

.846047 

.846175 

.846304 

.846432 

.846560 

.846688 

.846816 

9.846944 

.847071 

.847199 

.847327 

.847454 

.847582 

.847709 

.847836 

.847964 

.848091 

9.848218 

.848345 

.848472 

.848599 

.848726 

.848852 

.848979 

.849106 

.849232 

.849359 

.849485 

2.18 

2.18 

2.18 

2.18 

2.17 

2.17 

2.17 

2.17 

2.17 

2.17 

2.17 

2.17 

2.16 

2.16 

2.16 

2.16 

2.16 

2 16 
2.16 
2.16 

2T5 

2.15 

2.15 

2.15 

2.15 

2.15 

2.15 

2.15 

2.14 

2.14 

2.14 

2.14 

2.14 

2.14 

2.14 

2.14 

2.13 

2.13 

2.13 

2.13 

2.13 

2.13 

2.13 

2.13 

2.12 

2.12 

2.12 

2.12 

2.12 

2.12 

2.12 

2.12 

2.11 

2.11 

2.11 

2.11 

2.11 

2.11 

2.11 

2.11 

9.856934 

.856812 

.856690 

.856568 

.856446 

.856323 

.856201 

.856078 

.855956 

.855833 

9.855711 
.855588 
.855465 
.855342 
.855219 
.855096 
.854973 
.854850 
.854727 
.854603 

9.854480 

.854356 

.854233 

.854109 

.853986 

.853862 

.853738 

.853614 

.853490 

.853366 

9.853242 

.853118 

.852994 

.852869 

.852745 

.852020 

.852496 

.852371 

.852247 

.852122 

9.851997 

.851872 

.851747 

.851622 

.851497 

.851372 

.851246 

.851121 

.850996 

.850870 

9.850745 

.850619 

.850493 

.850308 

.850242 

.850116 

.849990 

.849864 

.849738 

.849611 

.849485 

2.03 

2.04 

2.04 

2.04 

2.04 

2.04 

2.04 

2.04 

2.04 

2.04 

2.05 

2.05 

2.05 

2.05 

2.05 

2.05 

2.05 

2.05 

2.06 

2.06 

2.06 

2.06 

2.06 

2.06 

2.06 

2.06 

2.06 

2.07 

2.07 

2.07 

2.07 

2.07 

2.07 

2.07 

2.07 

2.08 

2.08 

2.4)8 

2.08 

2.08 

2.08 

2.08 

2.08 

2.09 

2.09 

2.09 

2.09 

2.09 

2.09 

2.09 

2.09 

2.10 

2.10 

2.10 

2.10 

2.10 

2.10 

2.10 

2.10 

2.11 

9.984837 

.985090 

.985343 

.985596 

.985848 

.986101 

.986354 

.986607 

.986860 

.987112 

0.987365 

.987618 

.987871 

.988123 

.988376 

.988629 

.988882 

.989134 

.989387 

.989640 

9.989893 

.990145 

.990398 

.990651 

.990903 

.991156 

.991409 

.991662 

.991914 

.992167 

9.992420 

.992672 

.992925 

.993178 

.993430 

.993683 

.993936 

.994189 

.994441 

.994694 

9.994947 
.995199 
.995452 
.995705 
.995957 
.996210 
.996463 
.996715 
.996968 
.997221 

9.997473 

.997726 

.997979 

.998231 

.998484 

.998737 

.998989 

.999242 

.999495 

.999748 

10.000000 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4 21 
4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

4.21 

10.015163 

.014910 

.014657 

.014404 

.014152 

.013899 

.013646 

.013393 

.013140 

.012888 

10.012635 

.012382 

.012129 

.011877 

.011624 

.011371 

.011118 

.010866 

.010613 

.010360 

10.010107 

.009855 

.009602 

.009349 

.009097 

.008844 

.008591 

.008338 

.008086 

.007833 

10.007580 

.007328 

.007075 

.006822 

.006570 

.006317 

.006064 

.005811 

.005559 

.005306 

10.005053 

.004801 

.004548 

.004295 

.004043 

.003790 

.00:3537 

.003285 

.003032 

.002779 

10.002527 

.002274 

.002021 

.001769 

.001516 

.001263 

.001011 

.000758 

.000505 

.000252 

.000000 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 

M. 

Cosine. 

D. 1 

Sine. 

I). 1". 

Cotang. 

D. 1”. 

Tang. 

M. 































TABLE V. 


LATITUDES AND DEPARTURES, 

OR 

TRAVERSE TABLE. 


31 


[ 85 ] 



TABLE Y. TRAVERSE TABLE. 


8(3 


B’ng 

Ilist. 1. 

Disl. 3. 

Dist. 3. 

l>ist 4. 

IHst. 5 

B’ng 

O 1 

Lilt. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat 

Dep. 

Lat. 

Dep. 

« / 

0 15 

1.00(H) 

0.0044 

2.0000 0.0087 

3.0000 

0.0131 

4.0000 

0.0175 

5.0000 

0.0218 

89 45 

30 

0000 

0087 

1.9999 

0175 

2.9999 

0202 

3.9998 

0349 

4.9998 

0430 

30 

45 

0.9999 

0131 

9998 

0202 

9997 

0393 

9997 

0524 

9996 

0054 

15 

1 (V 

9998 

0175 

9997 

0349 

9995 

0524 

99! )4 

0098 

9992 

0873 

89 0 

15 

9998 

0218 

9995 

0430 

9993 

0054 

9990 

0873 

9988 

1091 

45 

30 

9997 

0202 

9993 

0524 

99! H) 

0785 

9980 

1047 

9983 

1309 

30 

45 

9995 

0305 

9991 

0011 

9980 

0910 

9981 

1222 

9977 

1527 

15 

2 0 

9994 

0349 

9988 

0098 

9982 

1047 

9970 

1396 

9970 

1745 

88 0 

15 

9992 

0393 

9985 

0785 

9977 

1178 

9909 

1570 

9901 

1963 

45 

30 

9990 

0430 

9981 

0872 

9971 

1309 

9902 

1745 

9952 

2181 

30 

45 

0.9988 

0.0480 

1.9977 

0.0900 

2.9905 

0.1439 

3.9954 

0.1919 

4.9942 

0.2399 

15 

3 0 

9980 

0523 

9973 

1047 

9959 

1570 

9!>45 

2093 

9931 

2617 

87 0 

15 

9984 

0507 

9908 

1134 

9952 

1701 

9936 

2208 

9920 

2835 

45 

30 

9981 

0010 

9903 

1221 

9944 

1831 

9925 

2442 

9907 

3052 

30 

45 

9979 

0054 

9957 

1308 

9930 

1902 

9914 

2610 

9893 

3270 

15 

4 0 

9970 

0098 

9951 

1395 

9927 

2093 

9903 

2790 

9878 

3488 

86 0 

15 

9973 

0741 

9945 

1482 

9918 

2223 

9890 

2964 

9803 

3705 

45 

30 

9909 

0785 

9938 

1509 

9908 

2354 

9877 

3138 

9846 

3923 

30 

45 

9900 

0828 

9931 

1G5G 

9897 

2484 

9803 

3312 

9828 

4140 

15 

5 0 

9902 

0872 

9924 

1743 

9880 

2015 

9848 

3480 

9810 

4358 

85 0 

15 

0.9958 

0.0915 

1.9910 

0.1830 

2.9874 

0.2745 

3.9832 

0.3060 

4.9790 

0.4575 

45 

30 

9954 

0958 

9908 

1917 

9802 

2875 

9810 

3834 

9770 

4792 

30 

45 

9950 

1002 

9899 

2004 

9849 

3000 

9799 

4008 

9748 

5009 

15 

6 0 

9945 

1045 

9890 

2091 

9830 

3130 

9781 

4181 

9720 

5220 

84 0 

15 

9941 

1089 

9881 

2177 

9822 

3200 

9702 

4355 

9703 

5443 

45 

30 

9930 

1132 

9871 

2204 

9807 

3390 

9743 

4528 

9679 

5600 

30 

45 

9931 

1175 

98G1 

2351 

9792 

3520 

9723 

4701 

9653 

5877 

15 

7 0 

9925 

1219 

9851 

2437 

9770 

3050 

9702 

4875 

9627 

6093 

83 0 

15 

9920 

1202 

9840 

2524 

9700 

3780 

9080 

5048 

9000 

6310 

45 

30 

9914 

1305 

9829 

2011 

9743 

3910 

9658 

5221 

9572 

0526 

30 

45 

0.9909 

0.1349 

1.9817 

0.2097 

2.9720 

0.4040 

3.9635 

0.5394 

4.9543 

0.0743 

15 

8 0 

9903 

1392 

9805 

2783 

9708 

4175 

9011 

5567 

9513 

6959 

82 0 

15 

9897 

1435 

9793 

2870 

9090 

4305 

9580 

5740 

9483 

7175 

45 

30 

9890 

1478 

9780 

2950 

9070 

4434 

9501 

5912 

9451 

7390 

30 

45 

9884 

1521 

9707 

3042 

9G51 

4504 

9534 

0085 

9418 

7000 

15 

9 0 

9877 

1504 

9754 

3129 

9031 

4693 

9508 

6257 

9384 

7822 

81 0 

15 

9870 

1007 

9740 

3215 

9010 

4822 

9480 

0430 

9350 

8037 

45 

30 

9803 

1050 

9720 

3301 

9589 

4951 

9451 

0002 

9314 

8252 

30 

45 

9850 

1093 

9711 

3387 

9507 

5080 

9422 

6774 

9278 

8407 

15 

10 0 

9848 

1730 

9090 

3473 

9544 

5209 

9392 

0940 

9240 

8682 

80 0 

15 

0.9840 

0.1779 

1.9081 

0.3559 

2.9521 

0.5338 

3.9302 

0.7118 

4.9202 

0.8897 

45 

30 

9833 

1822 

9005 

3045 

9498 

5407 

9330 

7289 

9163 

9112 

30 

45 

9825 

1805 

9049 

3730 

9474 

5596 

9298 

7461 

9123 

9326 

15 

11 0 

9810 

1908 

9633 

3810 

9449 

5724 

9205 

7632 

9081 

9540 

79 0 

15 

9808 

1951 

9010 

3902 

9424 

5853 

9231 

7804 

9039 

9755 

45 

30 

9799 

1994 

9598 

3987 

9398 

5981 

9197 

7975 

8990 

9968 

30 

45 

9790 

2030 

9581 

4073 

9371 

6109 

9102 

8140 

8952 

1.0182 

15 

12 0 

9781 

2079 

9503 

4158 

9344 

6237 

9120 

8310 

8907 

0396 

78 0 

15 

9772 

2122 

9545 

4244 

9317 

0365 

9089 

8487 

8802 

0009 

45 

30 

9703 

2104 

9520 

4329 

9289 

0493 

9052 

8058 

8815 

0822 

30 

45 

0.9753 

0.2207 

1.9507 

0.4414 

2.9200 

0.0021 

3.9014 

0.8828 

4.8767 

1.1035 

15 

13 0 

9744 

2250 

9487 

4499 

9231 

0749 

8975 

8998 

8719 

1248 

77 0 

15 

9734 

2292 

9408 

4584 

9201 

0870 

8935 

9108 

8609 

14(H) 

45 

30 

9724 

2&34 

9447 

4009 

9171 

7003 

8895 

9338 

8018 

1672 

30 

45 

9713 

2377 

9427 

4754 

9140 

7131 

8854 

9507 

8507 

1884 

15 

14 0 

9703 

2419 

9400 

4838 

9109 

7258 

8812 

9677 

8515 

2090 

76 0 

15 

9092 

2402 

9385 

4923 

9077 

7385 

8709 

9840 

8462 

2308 

45 

30 

9081 

2504 

9303 

5008 

9044 

7511 

8726 

1.0015 

8407 

2519 

30 

45 

9070 

2540 

9341 

5092 

9011 

7038 

8682 

0184 

8352 

2730 

15 

15 0 

9059 

2588 

9319 

5170 

8978 

7765 

8637 

0353 

8296 

2941 

75 0 

o t 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lilt. 

Dep. 

Lat. 

O f 

B’ng 

l>ist. 1. 

I>ist, 2. 

Dist. 3. 

I>ist.. 4. 

Dist. 5. 

B’ng 









































































TABLE Y. TRAVERSE TABLE, 


87 


B’nj. 

Dist. G. 

Dist. 7. 

Dist. 8. 

Dist. 9. 

Dist. 10. 

B’ng 

O i 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

| Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

O t 

o if 

5.999£ 

0.0202 

6.9998 

0.030f 

7.999! 

0.0343 

8.9993 

10.0393 

9.9993 

0.0433 

89 45 

3( 

) 999£ 

0521 

9997 

0011 

9997 

r 0698 

9997 

078f 

9993 

0873 

30 

4 ; 

999E 

078E 

9994 

091< 

! 9997 

1047 

9992 

1173 

9991 

1303 

15 

l ( 

9991 

1047 

998! 

1225 

9988 

1391 

998C 

1571 

9985 

1745 

>89 0 

if 

9981 

1308 

j98.': 

1527 

9981 

174E 

9973 

1903 

9973 

2181 

45 

3( 

997!; 

1571 

9971 

1835 

9973 

2094 

9903 

2353 

9900 

20D 

30 

4f 

9972 

1832 

9907 

2138 

9903 

2443 

9958 

2748 

9953 

3051 

15 

2 ( 

9963 

2094 

9957 

2443 

9951 

2792 

9945 

3141 

9939 

3493 

88 0 

If 

9951 

235G 

994C 

274£ 

9938 

3141 

9931 

3533 

9923 

3923 

45 

3C 

9943 

2017 

9933 

3053 

9924 

3490 

9914 

3920 

9905 

4362 

30 

4f 

5.9931 

0.2879 

0.9918 

0.3358 

7.9908 

0.3838 

8.9890 

0.4318 

9.9885 

0.4798 

15 

3 C 

9918 

3140 

9904 

3004 

9890 

4187 

9877 

4710 

9803 

5234 

87 0 

IE 

9904 

3402 

9887 

3908 

9871 

4535 

9855 

5102 

9839 

5009 

45 

3C 

9888 

3003 

9809 

4273 

9851 

4884 

9832 

5494 

9813 

6105 

30 

4E 

9872 

3924 

9850 

4578 

9829 

5232 

9807 

5880 

9780 

0540 

15 

4 0 

9854 

4185 

9829 

4883 

9805 

5581 

9781 

0278 

9750 

0970 

80 0 

15 

9835 

4447 

9808 

5188 

9780 

5929 

9753 

0070 

9725 

7411 

45 

30 

9815 

4708 

9784 

5492 

9753 

6277 

9723 

7001 

9092 

7840 

30 

45 

9794 

4908 

9700 

5797 

9725 

0025 

9091 

7453 

9057 

8281 

15 

5 0 

9772 

5229 

9734 

0101 

9090 

0972 

9058 

7844 

9019 

8710 

85 0 

15 

5.9748 

0.5490 

0.9700 

0.0405 

7.9004 

0.7320 

8.9022 

0 8235 

9.9580 

0.9150 

45 

30 

9724 

5751 

9078 

0709 

9632 

7008 

9586 

8020 

9540 

9585 

30 

45 

9698 

0011 

9048 

7013 

9597 

8015 

9547 

9017 

9497 

1.0019 

15 

6 0 

9071 

0272 

9017 

7317 

9502 

8302 

9507 

9408 

9452 

0453 

84 0 

15 

9043 

0532 

9584 

7021 

9525 

8709 

9465 

9798 

9400 

0887 

45 

30 

9014 

0792 

9550 

7924 

9480 

9050 

31421 

1.0188 

9357 

1320 

30 

45 

9584 

7052 

9515 

8228 

9445 

9403 

9376 

0578 

9307 

1754 

15 

7 0 

9553 

7312 

9478 

8531 

9404 

9750 

9329 

0908 

9255 

2187 

83 0 

15 

9520 

7572 

9440 

8834 

9300 

1.0090 

9280 

1358 

9200 

2020 

45 

30 

9487 

7832 

9401 

9137 

9310 

0442 

9230 

1747 

9144 

3053 

30 

45 

5.9452 

0.8091 

0.9301 

0.9440 

7.9209 

1.0788 

8.9178 

1.2137 

9.9087 

1.3485 

15 

8 0 

9416 

8350 

9319 

9742 

9221 

1134 

9124 

2520 

9027 

3917 

82 0 

. 15 

9379 

8010 

9270 

1.0044 

9172 

1479 

9009 

2914 

8905 

4349 

45 

30 

9341 

8809 

9231 

0347 

9121 

1825 

9011 

3303 

8902 

4781 

30 

45 

9302 

9127 

9185 

0049 

9069 

2170 

8953 

3091 

8830 

5212 

15 

0 0 

9201 

9380 

9138 

0950 

9015 

2515 

8892 

4079 

8709 

5043 

81 0 

15 

9220 

9045 

9090 

.1252 

8900 

2859 

8830 

4407 

8700 

0074 

45 

30 

9177 

9903 

9040 

1553 

8903 

3204 

8706 

4854 

8029 

6505 

30 

45 

9133 

1.0101 

8989 

1854 

8844 

3548 

8700 

5241 

8550 

6935 

15 

10 0 

9088 

0419 

8937 

2155 

8785 

3892 

8033 

5028 

8481 

7305 

80 0 

15 

5.9042 

1.0077 

0.8883 

1.2450 

7.8723 

1.4235 

8.8564 

1.6015 

9.8404 

1.7794 

45 

30 

8995 

0934 

8828 

2750 

8000 

4579 

8493 

0401 

8325 

8224 

30 

45 

8947 

1191 

8772 

3057 

8590 

4922 

8421 

6787 

8245 

8052 

15 

11 0 

8898 

1449 

8714 

3357 

8530 

5205 

8340 

7173 

8103 

9081 

79 0 

15 

8847 

1705 

8055 

3056 

8403 

5007 

8271 

7558 

8079 

9509 

45 

30 

8795 

1902 

8595 

3950 

8394 

5949 

8193 

7943 

7992 

9937 

30 

45 

8743 

2219 

8533 

4255 

8324 

0291 

8114 

8328 

7905 

2.0304 

15 

12 0 

8089 

2475 

8470 

4554 

8252 

6633 

8033 

8712 

7815 

0791 

78 0 

15 

8034 

2731 

8400 

4852 

8178 

0974 

7951 

9090 

7723 

1218 

45 

30 

8578 

2980 

8341 

5151 

8104 

7315 

7807 

9480 

7030 

1044 

30 

45 

5.8521 

1.3242 

S.8274 

1.5449 

7.8027 

1.7056 

8.7781 

1.9863 

9.7534 

2.2070 

15 

13 0 

8402 

3497 

8200 

5747 

7950 

7990 

7093 

2.0246 

7437 

2495 

77 0 

15 

8403 

3752 

8137 

0044 

7870 

8330 

7004 

0028 

7338 

2920 

45 

30 

8342 

4(X)7 

8000 

0341 

7790 

8070 

7513 

1010 

7237 

3345 

30 

45 

8281 

4201 

7994 

0038 

7707 

9015 

7421 

1392 

7134 

3709 

15 

14 0 

8218 

4515 

7921 

6935 

7024 

9354 

7327 

1773 

7030 

4192 

70 0 

15 

8154 

4709 

7840 

7231 

7538 

9092 

7231 

2154 

0923 

4015 

45 

30 

8089 

5023 

7770 

7527 

7452 

2.0030 

7133 

2534 

6815 

5038 

30 

45 

8023 

5270 

7093 

7822 

7304 

0308 

7034 

2914 

0705 

5400 

15 

15 0 

7950 

5529 

7015 

8117 

7274 

0700 

6933 

3294 

6593 

5882 

75 0 

0 t 

Dep. 

Lat. 

Dep. 

Lilt. 

Dep. 

T-Jiiti. 

Dep. 

Lat. 

Dep. 

Lat* 

o > 

B’ng 

Dist 

G. 

Dist 

7. 

Dist 

. 8. 

Dist 

9. 

nisi. 

10. 

B’ng 

















































































































88 TABLE Y. TRAVERSE TABLE. 


B’ng 1 

Dist 

. 1. 

Dist 

. 2. 

Dist. 3. 

• 

Dist. 1. 

Dist. 5. 

B’ng 

O 1 

Lat. 

Dep. 

Lat. | 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

O 1 

15 15 

0.9048 

0.2030 

1.9290 

0.5201 

2.8944 

0.7891 

3.8591 1.0521 

4.8239 

1.3152 

74 45 

30 

9030 

2072 

9273 

5345 

8909 

8017 

8545 

0090 

8182 

3302 

30 

45 

9025 

2714 

9249 

5429 

8874 

8143 

8498 

0858 

8123 

3572 

15 

10 0 

9013 

2750 

9225 

5513 

8838 

8269 

8450 

1025 

8003 

3782 

74 0 

15 

9000 

2798 

9201 

5597 

8801 

8395 

8402 

1193 

8002 

3991 

45 

30 

9588 

2840' 

9170 

5680 

8705 

8520 

8353 

1301 

7941 

4201 

30 

45 

9570 

2882 

9151 

5704 

8727 

8040 

8303 

1528 

7879 

4410 

15 

17 0 

9503 

2924 

9120 

5847 

8089 

8771 

8252 

1095 

7815 

4019 

73 0 

15 

9550 

2905 

9KH) 

5931 

8051 

8896 

8201 

1802 

7751 

4827 

45 

30 

9537 

3007 

9074 

0014 

8012 

9021 

8149 

2028 

7086 

5035 

30 

45 

0.9524 

0.3049 

1.9048 

0.0097 

2.8572 

0.9140 

3.8096 

1.2195 

4.7620 

1.5243 

15 

18 0 

9511 

3090 

9021 

0180 

8532 

9271 

8042 

2301 

7553 

5451 

72 0 

15 

9497 

3132 

8994 

0203 

8491 

9395 

7988 

2527 

7485 

5658 

45 

30 

9483 

3173 

8900 

0340 

8450 

9519 

7933 

2092 

7410 

5805 

30 

45 

9409 

3214 

8939 

0429 

8408 

9043 

7877 

2858 

7347 

6072 

15 

19 0 

9455 

3250 

8910 

6511 

8300 

9707 

7821 

3023 

7276 

6278 

71 0 

15 

9441 

3297 

8882 

0594 

8323 

9891 

7704 

3188 

7204 

6485 

45 

30 

9420 

3338 

8853 

6070 

8279 

1.0014 

7706 

3352 

7132 

6090 

30 

45 

9412 

3379 

8824 

6758 

8235 

0138 

7047 

3517 

7059 

6890 

15 

20 0 

9397 

3420 

8794 

6840 

8191 

0201 

7588 

3681 

0985 

7101 

70 0 

15 

0.9382 

0.3401 

1.8704 

0.0922 

2.8140 

1.0384 

3.7528 

1.3845 

4.6910 

1.7300 

45 

1 30 

9307 

3502 

8733 

7004 

8100 

0500 

7407 

4008 

6834 

7510 

30 

45 

9351 

3543 

8703 

7080 

8054 

0629 

7405 

4172 

6757 

7715 

15 

21 0 

9330 

3584 

8672 

7107 

8007 

0751 

7343 

4335 

6679 

7918 

69 0 

15 

9320 

3024 

8040 

7249 

7900 

0873 

7280 

4498 

6600 

8122 

45 

30 

9304 

3005 

8008 

7330 

7913 

0995 

7217 

4600 

6521 

8325 

30 

45 

9288 

3700 

8570 

7411 

7804 

1117 

7152 

4822 

0440 

8528 

15 

22 0 

9272 

3740 

8544 

7492 

7810 

1238 

70S7 

4984 

6359 

8730 

68 0 

15 

9255 

3780 

8511 

7573 

7700 

1359 

7022 

5140 

6277 

8932 

45 

30 

9239 

3827 

8478 

7054 

7710 

1481 

6955 

5307 

6194 

9134 

30 

45 

0.9222 

0.3807 

1.8444 

0.7734 

2.7666 

1.1601 

3.6888 

1.5468 

4.0110 

1 . 9:130 

15 

23 0 

9205 

3907 

8410 

7815 

7015 

1722 

6820 

5629 

0025 

9537 

67 0 

15 

9188 

3947 

8370 

7895 

7504 

1842 

6752 

5790 

5940 

9737 

45 

30 

9171 

3987 

8341 

7975 

7512 

1962 

6682 

5950 

5853 

9937 

30 

45 

9153 

4027 

8300 

8055 

7459 

2082 

6012 

6110 

5700 

2.0137 

15 

24 0 

9135 

4007 

8271 

8135 

7400 

2202 

6542 

6209 

5677 

0337 

66 0 

15 

9118 

4107 

8235 

8214 

7353 

2322 

6470 

6429 

5588 

0536 

45 

30 

9100 

4147 

8199 

8294 

7299 

2441 

6398 

6588 

5498 

0735 

30, 

45 

9081 

4187 

8103 

8373 

7244 

2500 

6326 

6746 

5407 

0933 

15 

25 0 

9003 

4220 

8120 

8452 

7189 

2679 

6252 

6905 

5315 

1131 

65 0 

15 

0.9045 

0.4200 

1.8089 

0.8531 

2.7134 

1.2797 

3.6178 

1.7003 

4.5223 

2.1328 

45 

30 

9020 

4305 

8052 

8010 

7078 

2915 

0103 

7220 

5129 

1520 

30 

45 

9007 

4344 

8014 

8089 

7021 

3033 

6028 

7378 

5035 

1722 

15 

20 0 

8988 

4384 

7970 

8707 

0904 

3151 

5952 

7535 

4940 

1919 

64 0 

15 

8969 

4423 

7937 

8840 

6! MM! 

3209 

5875 

7092 

4844 

2114 

45 

30 

8949 

4402 

7899 

8924 

6848 

3386 

5797 

7848 

4747 

2310 

301 

45 

8930 

4501 

7800 

9002 

6789 

3503 

5719 

8004 

4049 

2505 

15 

27 0 

8910 

4540 

7820 

9080 

6730 

3020 

50)40 

8100 

4550 

2700 

63 0| 

15 

8890 

4579 

7780 

9157 

6071 

3736 

5501 

8315 

4451 

2894 

45 

30 

8870 

4017 

7740 

9235 

6010 

3852 

5480 

8470 

4351 

3087 

30 

45 

0.8850 

0.4050 

1.7700 

0.9312 

2.6550 

1.3908 

3.5400 

1.8625 

4.4249 

2.3281 

15 

28 0 

8829 

4095 

7059 

9389 

0488 

4084 

5318 

8779 

4147 

3474 

62 0 

15 

8809 

4733 

7018 

9400 

6427 

4200 

5236 

8933 

40-45 

366( 

45 

30 

8788 

4772 

7570 

9543 

6365 

4315 

5153 

9086 

3941 

3858 

30 

45 

8707 

4810 

7535 

9620 

6302 

4430 

5009 

9240 

3836 

4049 

15 

29 0 

874b 

4848 

7492 

9090 

6239 

4544 

4985 

9392 

3731 

4246 

(61 0 

15 

8725 

4880 

7450 

9772 

6175 

465S 

490(1 

9545 

362E 

4431 

45 

30 

8704 

4924 

7407 

9848 

6111 

4772 

4814 

9097 

3518 

4621 

30 

45 

8082 

4902 

7304 

9924 

604C 

488€ 

4728 

9849 

341( 

4811 

15 

30 0 

8000 

5000 

7321 

1.000(1 

5981 

5CKX 

4041 

2.0000 

3301 

500( 

60 0 

o t 

Dei). 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

1 Lat. 

• o 

B’nfi 

Dist. 1. 

Dist. 2. 

Dist. 3. 

Dist. 4. 

Dist. 5, 

B’ng 
































































































TABLE V. TRAVERSE TABLE 


89 


B’ng 

Dist. 6. 

Dist. 7. 

Dist. 8 . 

Dist. 9. 

Dist. lO. 

iB’ng 

O i 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

1 Dep. 

Lat. 

1 Dep. 

Lat. 

Dep. 

O 1 

15 15 

5.7887 

1.5782 

6.7535 

1.8412 

7.7183 

2.1042 

8.6831 

2.3673 

9.6479 

2.6303 

74 45 

30 

7818 

6034 

7454 

8707 

7090 

1379 

6727 

4051 

6363 

6724 

30 

45 

7747 

6286 

7372 

9001 

6996 

1715 

6621 

4430 

6246 

7144 

15 

16 0 

7676 

6538 

7288 

9295 

6901 

2051 

6514 

4807 

6126 

7564 

74 0 

15 

7603 

6790 

7203 

9588 

6804 

2386 

6404 

5185 

6005 

7983 

45 

30 

7529 

7041 

7117 

9881 

6706 

2721 

6294 

5561 

5882 

8402 

30 

45 

7454 

7292 

7030 

2.01741 6606 

3056 

6181 

5938 

5757 

8820 

15 

17 0 

7378 

7542 

6941 

0466 

6504 

3390 

6067 

6313 

5630 

9237 

73 0 

15 

7301 

7792 

6851 

0758 

6402 

3723 

5952 

6689 

5502 

9654 

45 

30 

7223 

8042 

6760 

1049 

6297 

4056 

5835 

7064 

5372 

3.0071 

30 

45 

5.7144 

1.8292 

6.6668 

2.1341 

7.6192 

2.4389 

8.5716 

2.7438 

9.5240 

3.0486 

15 

18 0 

7063 

8541 

6574 

1631 

6085 

4721 

5595 

7812 

5106 

0902 

72 0 

15 

6982 

8790 

6479 

1921 

5976 

5053 

5473 

8185 

4970 

1316 

45 

30 

6899 

9038 

6383 

2211 

5866 

5384 

5349 

8557 

4832 

1730 

30 

45 

6816 

9286 

6285 

2501 

5754 

5715 

5224 

8930 

4693 

2144 

15 

19 0 

6731 

9534 

6186 

2790 

5641 

6045 

5097 

9301 

4552 

2557 

71 0 

15 

6645 

9781 

6086 

3078 

5527 

6375 

4968 

9672 

4409 

2969 

45 

30 

6558 

2.0028 

5985 

3366 

5411 

6705 

4838 

3.0043 

4264 

3381 

30 

45 

6471 

0275 

5882 

3654 

5294 

7033 

4706 

0413 

4118 

3792 

15 

20 0 

6382 

0521 

5778 

3941 

5175 

7362 

4572 

0782 

3969 

4202 

70 0 

15 

5.6291 

2.0767 

6.5673 

2.4228 

7.5055 

2.7689 

8.4437 

3.1151 

9.3819 

3.4612 

45 

30 

6200 

1012 

5567 

4515 

4934 

8017 

4300 

1519 

3667 

5021 

30 

45 

6108 

1257 

5459 

4800 

4811 

8343 

4162 

1886 

3514 

5429 

15 

21 0 

6015 

1502 

5351 

5086 

4686 

8669 

4022 

2253 

3358 

5837 

69 0 

15 

5920 

1746 

5241 

5371 

4561 

8995 

3881 

2619 

3201 

6244 

45 

30 

5825 

1990 

5129 

5655 

4433 

9320 

3738 

2985 

3042 

6650 

30 

45 

5729 

2233 

5017 

5939 

4305 

9645 

3593 

3350 

2881 

7056 

15 

22 0 

5631 

2476 

4903 

6222 

4175 

9969 

3447 

3715 

2718 

7461 

68 0 

15 

5532 

2719 

4788 

6505 

4043 

3.0292 

3299 

4078 

2554 

7865 

45 

30 

5433 

2961 

4672 

6788 

3910 

0615 

3149 

4442 

2388 

8268 

30 

45 

5.5332 

2.3203 

6.4554 

2.7070 

7.3776 

3.0937 

8.2998 

3.4804 

9.2220 

3.8671 

15 

23 0 

5230 

3444 

4435 

7351 

3640 

1258 

2845 

5166 

2050 

9073 

67 0 

15 

5127 

3685 

4315 

7632 

3503 

1580 

2691 

5527 

1879 

9474 

45 

30 

5024 

3925 

4194 

7912 

3365 

1900 

2535 

5887 

1706 

9875 

30 

45 

4919 

4165 

4072 

8192 

3225 

2220 

2378 

6247 

1531 

4.0275 

15 

24 0 

4813 

4404 

3948 

8472 

3084 

2539 

2219 

6606 

1355 

0674 

66 0 

15 

4706 

4643 

3823 

8750 

2941 

2858 

2059 

6965 

1176 

1072 

45 

30 

4598 

4882 

3697 

9029 

2797 

3175 

1897 

7322 

0996 

1469 

30 

45 

4489 

5120 

3570 

9306 

2651 

3493 

1733 

7679 

0814 

1866 

15 

25 0 

4378 

5357 

3442 

9583 

2505 

3809 

1568 

8036 

0631 

2262 

65 0 

15 

5.4267 

2.5594 

6.3312 

2.9860 

7.2356 

3.4125 

8.1401 

3.8391 

9.0446 

4.2657 

45 

30 

4155 

5831 

3181 

3.0136 

2207 

4441 

1233 

8746 

0259 

3051 

30 

45 

4042 

6067 

3049 

0411 

2056 

4756 

1063 

9100 

0070 

3445 

15 

26 0 

3928 

6302 

2916 

0686 

1904 

5070 

0891 

9453 

8.9879 

3837 

64 0 

15 

3812 

6537 

2781 

0960 

1750 

5383 

0719 

9806 

9687 

4229 

45 

30 

3696 

6772 

2645 

1234 

1595 

5696 

0544 

4.0158 

9493 

4620 

30 

45 

3579 

7006 

2509 

1507 

1438 

6008 

0368 

0509 

9298 

5010 

15 

27 0 

3460 

7239 

2370 

1779 

1281 

6319 

0191 

0859 

9101 

5399 

63 0 

15 

3341 

7472 

2231 

2051 

1121 

6630 

00121 

1209 

8902 

5787 

45 

30 

3221 

7705 

2091 

2322 

0961 

6940 

CO 

00 

cs 

1557 

8701 

6175 

30 

45 

5.3099 

2.7937 

6.1949 

3.2593 

7.0799 

3.7249 

7.9649 

4.1905 

8.8499 

4.6561 

15 

2 i 0 

2977 

8168 

1806 

2863 

0636 

7558 

9465 

2252 

8295 

6947 

62 0 

15 

2853 

8399 

1662 

3132 

0471 

786C 

9280 

2599 

8089 

7332 

45 

30 

2729 

8630 

1517 

3401 

0305 

8173 

9094 

2944 

7882 

7716 

30 

45 

2604 

8859 

1371 

3669 

0138 

8479 

8905 

3289 

7673 

8099 

15 

29 0 

2477 

9089 

1223 

3937 

G.9970 

8785 

8716 

3633 

7462 

8481 

61 0 

15 

2350 

9317 

1075 

4203 

9800 

9090 

8525 

3976 

7250 

8862 

45 

30 

2221 

9545 

0925 

4470 

9628 

9394 

8332 

4318 

7036 

9242 

30 

45 

2092 

9773 

0774 

4735 

9456 

9697 

8138 

4659 

6820 

9622 

15 

30 0 

1962 

3.0000 

0622 

5000 

9282 

4.0000 

7942 

5000 

6603 

5.00(H) 

60 0 

O 1 

Dep. 

Lat. 

Dep. : 

Lat. 

Dep. 

Lat. 

Dep 

Lat. 

Dep. 

Lat* 

O f 

B’ng 

Dist. 6. 

Dist. 7. 

Dist. 8. 

Dist. 9. 

Dist.lO. 

B’ng 


Cf 





































































































90 


TABLE V. TRAVERSE TABLE 


B’ng 

Dist. l. 

Dist. 2. 

Dist. 3. 

Dist. 4. 

Dist. 5. 

B’ng 

O 1 

Lat 

Dep 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

O t 

30 15 

0.8638 

0.5038 

1.7277 

1.0075 

2.5915 

1.5113 

3.4553 

2.0151 

4.3192 

2.5189 

59 45 

30 

8610 

5075 

7233 

0151 

5849 

5226 

4465 

0302 

3081 

5377 

30 

45 

8504 

5113 

7188 

0226 

5782 

5339 

4370 

0452 

2970 

5565 

15 

31 0 

8572 

5150 

7142 

0301 

6715 

5451 

4287 

0002 

2858 

5752 

59 0 

15 

8549 

5188 

7098 

0375 

5647 

5503 

4196 

0751 

2746 

5939 

45 

30 

8520 

5225 

7053 

0450 

5579 

6675 

4106 

0900 

2632 

6125 

30 

45 

8504 

6262 

7007 

0524 

5511 

5786 

4014 

1049 

2518 

6311 

15 

32 0 

8480 

5299 

0901 

0598 

5441 

5898 

3922 

1197 

2402 

6496 

58 0 

15 

8457 

5330 

0915 

0072 

5372 

0008 

3829 

1345 

2286 

6681 

45 

30 

8434 

5373 

6868 

0740 

5302 

6119 

3736 

1492 

2170 

6865 

30 

45 

0.8410 

0.5410 

1.0821 

1.0819 

2.5231 

1.0229 

3.3642 

2.1639 

4.2052 

2.7049 

15 

33 0 

8387 

5440 

0773 

0893 

5100 

0339 

3547 

1780 

1934 

7232 

57 0 

15 

8303 

5483 

0726 

0900 

5089 

0449 

3451 

1932 

1814 

7415 

45 

30 

8339 

5519 

0678 

1039 

5017 

6558 

3355 

2077 

1694 

7597 

30 

45 

8315 

5556 

6629 

1111 

4944 

0667 

3259 

2223 

1573 

7779 

15 

34 0 

8290 

5592 

6581 

1184 

4871 

6776 

3102 

2368 

1452 

7960 

50 0 

15 

8266 

5628 

0532 

1256 

4798 

6884 

3004 

2512 

1329; 

8140 

45 

30 

8241 

5604 

6483 

1328 

4724 

6992 

2965 

2656 

1200 

8320 

30 

45 

8210 

5700 

6433 

1400 

4649 

71(H) 

2866 

28(H) 

1082 

8500 

15 

35 0 

8192 

5736 

6383 

1472 

4575 

7207 

2706 

2943 

0958 

8679 

55 0 

15 

0.8100 

0.5771 

1.0333 

1.1543 

2.4499 

1.7314 

3.2666 

2.3080 

4.0832 

2.8857 

45 

30 

8141 

5807 

0282 

1014 

4423 

7421 

2565 

3228 

0700 

9035 

30 

45 

8110 

5842 

0231 

1685 

4347 

7527 

2403 

3370 

0579 

9212 

15 

30 0 

8090 

5878 

0180 

1750 

4271 

7634 

2301 

3511 

0451 

9389 

54 0 

15 

8064 

5913 

0129 

1826 

4193 

7739 

2258 

3052 

0322 

9505 

45 

30 

8039 

5948 

0077 

1890 

4110 

7845 

2154 

3793 

0193 

9741 

30 

45 

MO 13 

5983 

0025 

1906 

4038 

7950 

2050 

3933 

0063 

9916 

15 

37 0 

7980 

0018 

5973 

2030 

3959 

8054 

1945 

4073 

3.9932 

3.0091 

53 0 

15 

7900 

6053 

5920 

2106 

3880 

8159 

1840 

4212 

9800 

0205 

45 

30 

7934 

6088 

5807 

2175 

3801 

8203 

1734 

4350 

9608 

0438 

30 

45 

0.7007 

0.6122 

1.5814 

1.2244 

2.3721 

1.8307 

3.1628 

2.4489 

3.9534 

3.0611 

15 

38 0 

7880 

0157 

5700 

2313 

3040 

8470 

1520 

4620 

9400 

0783 

52 0 

15 

7853 

0191 

5700 

2382 

3560 

8573 

1413 

4764 

9266 

0955 

45 

30 

7820 

6225 

5652 

2450 

3478 

8675 

1304 

4901 

9130 

1120 

30 

45 

7700 

0259 

5598 

2518 

3397 

8778 

1195 

5037 

8994 

1296 

15 

30 0 

7771 

6293 

5543 

2580 

3314 

8880 

1086 

5173 

8857 

1466 

51 0 

15 

7744 

0327 

5488 

2654 

3232 

8981 

0976 

5308 

8720 

1635 

45 

30 

7710 

0301 

5432 

2722 

3149 

9082 

0865 

5443 

8581 

1804 

30 

45 

7688 

6394 

6377 

2789 

3065 

9183 

0754 

5578 

8442 

1972 

15 

40 0 

7060 

6428 

5321 

2856 

2981 

9284 

0642 

5712 

8302 

2139 

50 0 

15 

0.7032 

0.0401 

1.5265 

1.2922 

2.2897 

1.938413.0529 

2.5845 

3.8162 

3.2306 

45 

30 

7004 

0494 

5208 

2989 

2812 

9483 

0416 

5978 

8020 

2472 

30 

45 

7576 

0528 

5151 


2727 

9583 

0303 

6110 

7878 

2638 

15 

41 0 

7547 

0501 

5094 

3121 

2041 

9682 

0188 

6242 

7735 

2803 

49 0 

15 

7518 

0593 

5037 

3187 

2555 

9780 

0074 

0374 

7592 

2967 

45 

30 

7490 

0026 

4979 

3252 

2409 

9879 

2.9958 

6505 

7448 

3131 

30 

45 

7401 

6059 

4921 

3318 

2382 

9970 

9842 

6635 

7303 

3294 

15 

42 0 

7431 

6691 

4803 

3383 

2294 

2.0074 

9726 

0765 

7157 

3457 

48 0 

15 

7402 

6724 

4804 

3447 

2207 

0171 

9609 

6895 

7011 

3618 

45 

30 

7373 

6756 

4746 

3512 

2118 

0268 

9491 

7024 

6864 

3780 

30 

45 

0.7343 

0.6788 

1.4686 

1.3570 

2.2030 

2.0364 

2.9373 

2.7152 

3.6716 

3.3940 

15 

43 0 

7314 

0820 

4627 

3040 

1941 

0400 

9254 

7280 

6568 

4100 

47 0 

15 

7284 

6852 

4507 

3704 

1851 

0555 

9135 

7407 

6419 

4259 

45 

30 

7254 

6884 

4507 

3767 

1761 

0651 

9015 

7534 

6269 

4418 

30 

45 

7224 

6915 

4447 

3830 

1071 

0745 

8895 

7661 

6118 

4576 

15 

44 0 

7193 

6947 

4387 

3893 

1580 

0840 

8774 

7786 

5967 

4733 

46 0 

15 

7103 

6978 

4320 

3956 

1489 

0934 

8652 

7912 

5815 

4890 

45 

30 

7133 

7009 

4205 

4018 

1398 

1027 

8530 

8036 

5663 

5045 

30 

45 

7102 

7040 

4204 

4080 

1306 

1120 

8407 

8161 

5509 

5201 

15 

45 0 

7071 

7071 

4142 

4142 

1213 

1213 

8284 

8284 

5355 

5355 

45 0 

0 t 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

o t 

B’ng 

Dist. 1. 

Dist. 2. 

Dist, 3. 

Dist. 4. 

Dist. 5. 

H 





























































































TABLE Y. TRAVERSE TABLE. 91 


B 


Dist. 6. 

Dist. 7. 

Dist. 8. 

Dist. 9. 

Dist. 10. 

B’ng 

O ! 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

LSitt 

Dep. 

Lat. 

Dep. 

o i 

30 

15 

5.1830 

3.0220 

6.0468 

3.5264 

6.9107 

4.0302 

7.7745 

4.5340 

8.6384 

5.0377 

59 45 


3(1 

1698 

0452 

0314 

5528 

8930 

0603 

7547 

5678 

6163 

0754 


30 


45 

1564 

0678 

0158 

5791 

8753 

0903 

7347 

6016 

5941 

1129 


15 

31 

0 

1430 

0902 

0002 

6053 

8573 

1203 

7145 

6353 

5717 

1504 

59 

0 


15 

1295 

1126 

5.9844 

6314 

8393 

1502 

6942 

6690 

5491 

1877 


45 


30 

1158 

1350 

9685 

6575 

8211 

1800 

6738 

7025 

5264 

2250 


30 


45 

1021 

1573 

9525 

6835 

8028 

2097 

6532 

7359 

5035 

2621 


15 

32 

0 

0883 

1795 

9363 

7094 

7844 

2394 

6324 

7693 

4805 

2992 

58 

0 


15 

0744 

2017 

9201 

7353 

7658 

2689 

6116 

8025 

4573 

3361 


45 


30 

0603 

2238 

9037 

7611 

7471 

2984 

5905 

8357 

4339 

3730 


30 


45 

5.0462 

3.2458 

5.8873 

3.7868 

6.7283 

4.3278 

7.5694 

4.8688 

8.4104 

5.4097 


15 

33 

0 

0320 

2678 

8707 

8125 

7094 

3571 

5480 

9018 

3867 

4464 

57 

0 


15 

0177 

2898 

8540 

8381 

6903 

3863 

5266 

9346 

3629 

4829 


45 


30 

0033 

3116 

8372 

8636 

6711 

4155 

5050 

9674 

3389 

5194 


30 


45 

4.9888 

3334 

8203 

8890 

6518 

4446 

4832 

5.0001 

3147 

5557 


15 

34 

0 

9742 

3552 

8033 

9144 

6323 

4735 

4613 

0327 

2904 

5919 

56 

0 


15 

9595 

3768 

7861 

9396 

6127 

5024 

4393 

0652 

2659 

6280 


45 


30 

9448 

3984 

7689 

9648 

5930 

5312 

4171 

0977 

2413 

6641 


30 


45 

9299 

4200 

7515 

9900 

5732 

5600 

3948 

1300 

2165 

7000 


15 

35 

0 

9149 

4415 

7341 

4.0150 

5532 

5886 

3724 

1622 

1915 

7358 

55 

0 


15 

4.8998 

3.4629 

5.7165 

4.0400 

6.5331 

4.6172 

7.3498 

5.1943 

8.1664 

5.7715 


45 


30 

8847 

4842 

6988 

0649 

5129 

6456 

3270 

2263 

1412 

8070 


30 


45 

8694 

5055 

6810 

0897 

4926 

6740 

3042 

2582 

1157 

8425 


15 

36 

0 

8541 

5267 

6631 

1145 

4721 

7023 

2812 

2901 

0902 

8779 

54 

0 


15 

8387 

5479 

6451 

1392 

4516 

7305 

2580 

3218 

0644 

9131 


45 


30 

8231 

5689 

6270 

1638 

4309 

7586 

2347 

3534 

0386 

9482 


30 


45 

8075 

5899 

6088 

1883 

4100 

7866 

2113 

3849 

0125 

9832 


15 

37 

0 

7918 

6109 

5904 

2127 

3891 

8145 

1877 

4163 

7.9864 

6.0182 

53 

0 


15 

7760 

6318 

5720 

2371 

3680 

8424 

1640 

4476 

9600 

0529 


45 


30 

7601 

6526 

5535 

2613 

3468 

8701 

1402 

4789 

9335 

0876 


30 


45 

4.7441 

3.6733 

5.5348 

4.2855 

6.3255 

4.8977 

7.1162 

5.5100 

7.9069 

6 -1222 


15 

38 

0 

7281 

6940 

5161 

3096 

3041 

9253 

0921 

5410 

8801 

1566 

52 

0 


15 

7119 

7146 

4972 

3337 

2825 

9528 

0679 

5718 

8532 

1909 


45 


30 

6956 

7351 

4783 

3576 

2609 

9801 

0435 

6026 

8261 

2251 


30 


45 

6793 

7555 

4592 

3815 

2391 

5.0074 

0190 

6333 

7988 

2592 


15 

39 

0 

6629 

7759 

4400 

4052 

2172 

0346 

6.9943 

6639 

7715 

2932 

51 

0 


15 

6464 

7962 

4207 

4289 

1951 

0616 

9695 

6943 

7439 

3271 


45 


30 

6297 

8165 

4014 

4525 

1730 

0886 

9446 

7247 

7162 

3608 


30 


45 

6131 

8366 

3819 

4761 

1507 

1155 

9196 

7550 

6884 

3944 


15 

40 

0 

5963 

8567 

3623 

4995 

1284 

1423 

8944 

7851 

6604 

4279 

50 

0 


15 

4.5794 

3.8767 

5.3426 

4.5229 

6.1059 

5.1690 

6.8691 

5.8151 

7.6323 

6.4612 


45 


30 

5624 

8967 

3228 

5461 

0832 

1956 

8437 

8450 

6041 

4945 


30 


45 

5454 

9166 

3030 

5693 

0605 

2221 

8181 

8748 

5756 

5276 


15 

41 

0 

5283 

9364 

2830 

5924 

0377 

2485 

7924 

9045 

5471 

5606 

49 

0 


15 

5110 

9561 

2629 

6154 

0147 

2748 

7666 

9341 

5184 

5935 


45 


30 

4937 

9757 

2427 

6383 

5.9916 

3010 

7406 

9636 

4896 

6262 


30 


45 

4763 

9953 

2224 

6612 

9685 

3271 

7145 

9929 

4606 

6588 


15 

42 

0 

4589 

4.0148 

2020 

6839 

9452 

3530 

6883 

6.0222 

4314 

6913 

48 

0 


15 

4413 

0342 

1815 

7066 

9217 

3789 

6620 

0513 

4022 

7237 


45 


30 

4237 

0535 

1609 

7291 

8982 

4047 

6355 

0803 

3728 

7559 


30 


45 

4.4059 

4.0728 

5.1403 

4.7516 

5.8746 

5.4304 

6.6089 

6.1092 

7.3432 

6.7880 


15 

43 

0 

3881 

0920 

1195 

7740 

8508 

4560 

5822 

1380 

3135 

8200 

47 

0 


15 

3702 

1111 

0986 

7963 

8270 

4815 

5553 

1666 

2837 

8518 


45 


30 

3522 

1301 

0776 

8185 

8030 

5068 

5284 

1952 

2537 

8835 


30 


45 

3342 

1491 

0565 

8406 

7789 

5321 

5013 

2236 

2236 

9151 


15 

44 

0 

3160 

1680 

0354 

8626 

7547 

5573 

4741 

2519 

1934 

9466 

46 

0 


15 

2978 

1867 

0 U1 

8845 

7304 

5823 

4467 

2801 

1630 

9779 


45 


30 

2795 

2055 

4.9928 

9064 

7060 

6073 

4193 

3082 

1325 

7.0091 


30 


45 

2611 

2241 

9713 

9281 

6815 

6321 

3917 

3361 

1019 

0401 


15 

45 

0 

2426 

2426 

9497 

9497 

6569 

6569 

3640 

3640 

0711 

0711 

45 

0 

o * 

Dep. 

Lat. 

Dep. 

Lut. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

O 

t 

B’ng 

Hist. 6. 

Dist. 7. 

Dist, H. 

Dist. 9. 

Dist. 10. 

B’ng 






















































































92 


TABLE YI. DEPARTURES, 


For Correction of Courses on Random lAncs. 


Minutes. 

10 Chains. 

20 Chains 

• 

1 

.003 

.006 

2 

006 

012 

3 

009 

017 

4 

012 

023 

5 

014 

029 

6 

017 

035 

7 

020 

041 

8 

023 

046 

9 

026 

052 

10 

029 

058 

11 

032 

064 

12 

035 

070 

13 

038 

076 

14 

041 

081 

15 

044 

087 

16 

046 

093 

17 

049 

099 

18 

052 

105 

19 

055 

110 

20 

058 

116 

21 

061 

122 

22 

064 

128 

23 

067 

134 

24 

070 

140 

25 

073 

145 

26 

076 

151 

27 

078 

157 

28 

081 

163 

29 

084 

169 

30 

087 

174 

31 

090 

180 

32 

093 

186 

33 

096 

192 

34 

099 

198 

35 

102 

204 

36 

105 

209 

37 

108 

215 

38 

110 

221 

39 

113 

227 

40 

116 

233 

41 

119 

238 

42 

122 

244 

43 

125 

250 

44 

128 

256 

45 

131 

262 

46 

134 

268 

47 

137 

273 

48 

140 

279 

49 

142 

285 

50 

145 

291 

51 

148 

297 

52 

151 

302 

53 

154 

308 

54 

157 

314 

55 

160 

320 

56 

163 

326 

57 

166 

331 

58 

169 

337 

59 

172 

343 

60 

174 

349 


40 Chains. 

80 Chains. 

Minutes 

.012 

.023 

1 

023 

046 

2 

035 

070 

3 

046 

093 

4 

058 

116 

5 

070 

140 

6 

081 

163 

7 

093 

186 

8 

105 

209 

9 

116 

233 

10 

128 

256 

11 

140 

279 

12 

151 

302 

13 

163 

326 

14 

174 

349 

15 

186 

372 

16 

198 

396 

17 

209 

419 

18 

221 

442 

19 

233 

466 

20 

244 

488 

21 

256 

512 

22 ' 

268 

535 

23 

279 

668 

24 

291 

581 

25 

302 

605 

26 

314 

628 

27 

326 

651 

28 

337 

674 

29 

349 

698 

30 

361 

722 

31 

372 

744 

32 

384 

767 

33 

395 

790 

34 

407 

814 

35 

419 

837 

36 

430 

860 

37 

442 

883 

38 

454 

906 

39 

465 

929 

40 

477 

953 

41 

488 

976 

42 

500 

999 

43 

512 

1.022 

44 

523 

1.045 

45 

535 

1.068 

46 

546 

1.092 

47 

558 

1.115 

48 

570 

1.138 

49 

581 

1.161 

50 

593 

1.184 

51 

605 

1.207 

52 

616 

1.230 

53 

628 

1.253 

54 

639 

1.276 

55 

651 

1.299 

56 

663 

1.323 

57 

674 

1.346 

58 

686 

1.369 

59 

698 

1.392 

60 






















TABLE VII. NATURAL SECANTS 


93 


1 ° — 

- - - 11° 

11 ° - - 

- - - - 21 

21 * - - 

- - - *31° 

31‘ 

- - - - 41“ 

Angle. 

Secant. 

Angle. 

Secant. 

Angle. 

Secant. 

Angle. 

Secant. 

o • 

1 

1.00015 

O 1 

11 

1.01872 

o r 

21 

1.07115 

o 

31 

1.16663 

10 

1.00021 

10 

1.01930 

10 

1.07235 

10 

1.16868 

20 

1.00027 

20 

1.01989 

20 

1.07356 

20 

1.17075 

30 

1.00034 

30 

1.02049 

30 

1.07479 

30 

1.17283 

40 

1.00042 

40 

1.02110 

40 

1.07602 

40 

1.17493 

50 

1.00051 

50 

1.02171 

50 

1.07727 

50 

1.17704 

2 

1.00061 

12 

1.02234 

22 

1.07853 

32 

1.17918 

10 

1.00072 

10 

1.02298 

10 

1.07981 

10 

1.18133 

20 

1.00083 

20 

1.02362 

20 

1.08109 

20 

1.18350 

30 

1.00095 

30 

1.02428 

30 

1.08239 

30 

1.18569 

40 

1.00108 

40 

1.02494 

40 

1.08370 

40 

1.18790 

50 

1.00122 

50 

1.02562 

50 

1.08503 

50 

1.19012 

3 

1.00137 

13 

1.02630 

23 

1.08636 

33 

1.19236 

10 

1.00153 

10 

1.02700 

10 

1.08771 

10 

1.19463 

20 

1.00169 

20 

1.02770 

20 

1.08907 

20 

1.19691 

30 

1.00187 

30 

1.02842 

30 

1.09044 

30 

1.19920 

40 

1.00205 

40 

1.02914 

40 

1.09183 

40 

1.20152 

50 

1.00224 

50 

1.02987 

50 

1.09323 

60 

1.20386 

4 

1.00244 

14 

1.03061 

24 

1.09464 

34 

1.20622 

10 

1.00265 

10 

1.03137 

10 

1.09606 

10 

1.20859 

20 

1.00287 

20 

1.03213 

20 

1.09750 

20 

1.21099 

30 

1.00309 

30 

1.03290 

30 

1.09895 

30 

1.21341 

40 

1.00333 

40 

1.03368 

40 

1.10041 

40 

1.21584 

50 

1.00357 

50 

1.03447 

50 

1.10189 

50 

1.21830 

5 

1.00382 

15 

1.03528 

25 

1.10338 

35 

1.22070 

10 

1.00408 

10 

1.03609 

10 

1.10488 

10 

1.22327 

20 

1.00435 

20 

1.03691 

20 

1.10640 

20 

1.22579 

30 

1.00163 

30 

1.03774 

30 

1.10793 

30 

1.22833 

40 

1.00491 

40 

1.03858 

40 

1.10947 

40 

1.23089 

50 

1.00521 

50 

1.03944 

50 

1.11103 

50 

1.23347 

6 

1.00551 

16 

1.04030 

26 

1.11260 

36 

1.23607 

10 

1.00582 

10 

1.04117 

10 

1.11419 

10 

1.23869 

20 

1.00614 

20 

1.04206 

20 

1.11579 

20 

1.24134 

30 

1.00647 

30 

1.04295 

30 

1.11740 

30 

1.24400 

40 

1.00681 

40 

1.04385 

40 

1.11903 

40 

1.24669 

50 

1.00715 

50 

1.04477 

50 

1.12067 

50 

1.24940 

7 

1.00751 

17 

1.04569 

27 

1.12233 

37 

1.25214 

10 

1.00787 

10 

1.04663 

10 

1.12400 

10 

1.25489 

20 

1.00825 

20 

1.04757 

20 

1.12568 

20 

1.25767 

30 

1.00863 

30 

1.04853 

30 

1.12738 

30 

1 26047 

40 

1.00902 

40 

1.04950 

40 

1.12910 

40 

1 26330 

50 

1.00942 

50 

1.05047 

50 

1.13083 

50 

1.26615 

8 

1.00983 

18 

1.05146 

28 

1.13257 

38 

1.26902 

10 

1.01024 

10 

1.05246 

10 

1.13433 

10 

1.27191 

20 

1.01067 

20 

1.05347 

20 

1.13610 

20 

1.27483 

30 

1.01111 

30 

1.05449 

30 

1.13789 

30 

1.27778 

40 

1.01155 

40 

1.05552 

40 

1.13970 

40 

1.28075 

50 

1.01200 

50 

1.05657 

50 

1.14152 

50 

1.28374 

9 

1.01247 

19 

1.05762 

29 

1.14335 

39 

1.28676 

10 

1.01294 

10 

1.05869 

10 

1.14521 

10 

1.28980 

20 

1.01342 

20 

1.05976 

20 

1.14707 

20 

1.29287 

30 

1.01391 

30 

1.06085 

30 

1.14896 

30 

1.29597 

40 

1.01440 

40 

1.06195 

40 

1.15085 

40 

1.29909 

50 

1.01491 

50 

1.06306 

50 

1.15277 

50 

1.30223 

10 

1.01543 

20 

1.06418 

30 

1.15470 

40 

1.30541 

10 

1.01595 

10 

1.06531 

10 

1.15665 

10 

1.30831 

20 

1.01649 

20 

1.06645 

20 

1.15861 

20 

1.31183 

30 

1.01703 

30 

1.06761 

30 

1.16059 

30 

1.31509 

40 

1.01758 

40 

1.06878 

40 

1.16259 

40 

1.31837 

50 

1.01815 

50 

1.06995 

50 

1.16460 

50 | 

1.32168 








































94 TABLE VII. NATURAL SECANTS. 


41° - - 

- 46" 

46° - 

51° 

51° - 

5«° 

1 

1 

0 

IS 

- 61° 

Angle. 

Secant. 

Angle. 

Secant. 

Angle. 

Secant. 

Angle. 

Secant. 

O 1 

41 

1.32501 

o » 

46 

1.43956 

« t 

51 

1.58902 

O 9 

56 

1.78829 

10 

1.32838 

10 

1.44301 

10 

1.59475 

10 

1.79604 

20 

1.33177 

20 

1.44831 

20 

1.60054 

20 

1.80388 

30 

1.33510 

30 

1.45274 

30 

1.60639 

30 

1.81180 

40 

1.33864 

40 

1.45721 

40 

1.61229 

40 

1.81081 

50 

1.34212 

50 

1.46173 

50 

1.61825 

50 

1.82700 

42 

1.34563 

47 

1.46628 

52 

1.62427 

57 

1.83608 

10 

1.34017 

10 

1.47087 

10 

1.63035 

10 

1.84435 

20 

1.35274 

20 

1.47551 

20 

1.63648 

20 

1.85271 

30 

1.35634 

30 

1.48010 

30 

1.64268 

30 

1.86116 

40 

1.35907 

40 

1.48401 

40 

1.64804 

40 

1.86900 

50 

1.36363 

50 

1.48067 

50 

1.65526 

50 

1.87834 

43 

1.36733 

48 

1.49448 

53 

1.66164 

58 

1.88708 

10 

1.37105 

10 

1.49033 

10 

1.66809 

10 

1.80501 

20 

1.37481 

20 

1.50422 

20 

1.67460 

20 

1.90485 

30 

1.37860 

30 

1.50016 

30 

1.68117 

30 

1.91388 

40 

1.38242 

40 

1.51415 

40 

1.68782 

40 

1.92302 

50 

1.38628 

50 

1.51918 

50 

1.69452 

50 

1.93226 

44 

1.30016 

40 

1.52425 

54 

1.70130 

59 

1.94160 

10 

1.39400 

10 

1.52938 

10 

1.70815 

10 

1.95106 

20 

1.39804 

20 

1.53455 

20 

1.71506 

20 

1.96062 

30 

1.40203 

30 

1.53977 

30 

1.72205 

30 

1.97029 

40 

1.40606 

40 

1.54504 

40 

1.72911 

40 

1.98008 

50 

1.41012 

50 

1.55036 

60 

1.73624 

50 

1.9891)8 

45 

1.41421 

50 

1.55572 

55 

1.74345 

60 

2.00000 

10 

1.41835 

10 

1.56114 

10 

1.75073 

10 

2.01014 

20 

1.42251 

20 

1.56661 

20 

1.75808 

20 

2.02039 

30 

1.42670 

30 

1.57213 

30 

1.76552 

30 

2.03077 

40 

1.43096 

40 

1.57771 

40 

1.77303 

40 

2.04128 

50 

1.43524 

50 

1.58333 

50 

1.78062 

50 

2.05191 

— . ... — 


JAN. 1 . TABLE VIII. JAN. 1 . 

AZIMUTHS OP rOLARIS AT EXTREME ELONGATIONS. 


Lat. N. 

1896 

1897 

1898 

1899 

1900 

1901 

1902 

1903 

1904 

1905 

o 

0 / 

o / 

O ' 

o t 

O f 

O f 

o t 

o / 

o t 

O / 

25 

1 23.5 

l 22.2 

1 21 8 

1 21.5 

1 21.1 

1 20.8 

1 20.5 

1 20.1 

l 19.8 

1 19 4 

26 

23.2 

22.9 

22.5 

22.2 

21.8 

21.5 

21.1 

20.8 

20.4 

20.1 

27 

23.9 

23 6 

23.2 

22.9 

22.5 

22.2 

21.8 

21.5 

21.1 

20 8 

28 

24.7 

24.4 

24.0 

23.6 

23.3 

22.9 

22.6 

22.2 

21.9 

21.5 

29 

25.5 

25.2 

24.8 

24.4 

24.1 

23.7 

23.4 

23.0 

22.6 

22.3 

30 

26.4 

26 0 

25.6 

25.3 

24.9 

24.6 

24.2 

23.8 

23.5 

23 1 

31 

27.3 

26.9 

26 5 

26.2 

25.8 

25.4 

25.1 

24.7 

24.3 

24.0 

32 

28.2 

27.8 

27.5 

27.1 

26 7 

26.4 

26.0 

25.6 

25.2 

24 9 

33 

29 2 

28.8 

28.4 

28.1 

27.7 

27.3 

26.9 

26.6 

26.2 

25 8 

34 

30.2 

29.8 

29.5 

29.1 

28.7 

28.3 

27.2 

27.6 

27.2 

26.8 

35 

31.3 

30 9 

30.5 

30.2 

29 8 

29.4 

29.0 

28.6 

28.2 

27.8 

86 

32.5 

32 1 

31 7 

31.3 

30.9 

30.5 

30.1 

29.7 

29.3 

29.0 

37 

33.7 

33.3 

32.9 

32.5 

32.1 

31.7 

31.3 

30.9 

30.5 

30.1 

38 

34.9 

34.5 

34.1 

33.7 

33.3 

32.9 

32.5 

32.1 

31.7 

31.3 

39 

36.2 

35.8 

35 4 

35.0 

34.6 

34.2 

33.8 

33.4 

33.0 

82.6 

40 

37.6 

37.2 

36.8 

36.4 

36.0 

35.6 

35.2 

34.8 

34.4 

34.0 

41 

39 1 

38.7 

38.3 

37.9 

37.4 

87.0 

36.6 

36.2 

35.8 

35.4 

42 

40.6 

40.2 

39.8 

39.4 

39.0 

38.5 

38.1 

37.7 

373 

36.9 

43 

42.3 

41.8 

41.4 

41.0 

40.6 

40.1 

39.7 

39.3 

38.8 

38.4 

44 

44.0 

43.5 

43.1 

42.7 

42.2 

41.8 

41.4 

40.9 

40.5 

40.1 

45 

45 8 

45.3 

44.9 

44.4 

44.0 

43.6 

43.1 

42.7 

42.2 

41.8 

46 

47.7 

47.2 

46.8 

46.3 

45.9 

45.4 

45.0 

44.5 

44 0 

43.0 

47 

49.7 

49.2 

48.8 

48.3 

47.8 

47.4 

46 9 

46.5 

46.0 

45.5 

48 

51.8 

51.3 

50.8 

50.4 

49.9 

49.4 

49.0 

48.5 

48.0 

47 6 

49 

54.0 

53.5 

53.1 

52.6 

52 1 

51.6 

51.2 

50.7 

50.2 

49.7 

50 

1 56.4 

1 55.9 

1 55.4 

1 54.9 

1 54.4 

1 53.9 

1 58.4 

1 53.0 

1 52.5 

1 52.0 

































































95 


TABLE IX. MULTIPLIERS OF R, 


For one revolution of Grcidientcr Screw, used in finding d' and d. Page 99. 


Elevation. 

Multipliers 
of r. 

Elevation. 

Multipliers 
of r. 

Elevat’n. 

Multipliers 
of r. 



Inc. 

Ilor. 



Inc. 

Ilor. 



Inc. 

Ilor. 

e 


Dist. 

Dist. 

e. 


Dist. 

Dist. 

e. 

Dist. 

Dist. 

o 

1 

/ 

00 

99.97 

99.95 

0 

14 

/ 

96.79 

93.91 

o 

22 

/ 

30 

92.01 

85.01 

2 


99.90 

99.84 

14 

30 

96.56 

93.49 

23 


91.66 

84.37 

3 


99.81 

99.67 

15 


96.33 

93.05 

23 

30 

91.31 

83.73 

4 


99.69 

99.44 

15 

30 

96.09 

92.59 

24 


90.95 

83.08 

5 


99.53 

99.15 

16 


95.85 

92.13 

24 

30 

90.58 

82.42 

6 


99.35 

98.80 

16 

30 

95.00 

91.66 

25 


90.21 

81.75 

7 


99.13 

98.39 

17 


95 34 

91.17 

25 

30 

89.83 

81.08 

8 


98.89 

97.92 

17 

30 

95.07 

90.67 

26 


89.44 

80.39 

9 


98.61 

97.39 

18 


94.80 

90.15 

26 

30 

89.05 

79.69 

10 


98.31 

96.81 

18 

30 

94.52 

89.63 

27 


88.65 

78.99 

10 

30 

98.14 

96.50 

19 


94.23 

89.09 

27 

30 

88.24 

78.27 

11 


97.97 

96.17 

19 

30 

93.93 

88.54 

28 


87.83 

77.55 

11 

30 

97.79 

95.83 

20 


93.63 

87.97 

28 

30 

87.40 

76.81 

12 


97.61 

95.47 

20 

30 

93.32 

87.41 

29 


86.98 

76.07 

12 

30 

97.41 

95.10 

21 


93.00 

86.82 

29 

30 

86.54 

75.32 

13 


97.21 

94.72 

21 

30 

92.68 

86.23 

30 


86.10 

74.57 

13 

30 

97.00 

94.22 

22 


92.34 

85.61 

30 

30 

85.66 

73.81 


TABLE X. ANGLES OF ELEVATION, 


Corresponding to numbers of Revolution of the Gradienter Screw, 


Screw. 

Angle. 

l 

Screw. 

Angle. 

Screw. 

Angle. 

Rev. Div. 

o t // 

Rev. Div. 

o » ft 

Rev. Div. 

o t It 

0 1 

0 00 21 

0 10 

0 3 26 

1 00 

0 34 23 

2 

0 41 

20 

6 53 

2 

1 08 45 

3 

1 02 

30 

10 19 

3 

1 43 06 

4 

1 23 

40 

13 45 

4 

2 17 26 

5 

1 43 

50 

17 11 

5 

2 51 45 

6 

2 04 

60 

20 38 

6 

3 26 01 

7 

2 24 

70 

24 04 

7 

4 00 15 

8 

2 45 

80 

27 30 

8 

4 34 26 

9 

3 06 

90 

30 56 

9 

5 08 34 

0 io 

0 3 26 

1 00 

0 34 23 

10 00 

5 42 38 











































































90 TABLE XI. MEAN REFRACTIONS, 


In Declination, for use with Solar Compass. 


, 

Hour Angle. 

Declinations. 

For Latitude 30°. 

i20° 

+ 15° 

+ 10° 

+ 5 C 

0* 

—5° 

—10" 

—15° 

—20° 

oh. 

2 

3 

4 

6 

10" 

14 

20 

32 

100 

15" 

19 

26 

39 

110 

21" 

25 

32 

46 

1'24 

27" 

31 

39 

52 

1 '52 

33" 

38 

47 

1 06 

2 07 

40" 

46 

55 

119 

2 44 

48" 

54 

106 

1 35 

3 46 

57" 

105 

1 19 

1 57 

5 43 

ros" 

1 18 

1 36 

2 29 

13 06 

For Latitude 32° 30'. 

Oh. 

2 

3 

4 

5 

13" 

17 

23 

35 

1’03 

18" 

22 

29 

43 

1'15 

24" 

28 

35 

51 

1'31 

30" 

35 

43 

1 01 

1 53 

36” 

42 

51 

1'13 

2 20 

44" 

50 

l’Ol 

1 27 

3 05 

52" 

100 

1 13 

1 46 

4 25 

1'02" 

1 11 

1 28 

2 13 

7 36 

114" 

1 26 

1 47 

2 54 

For Latitude 35°. 

Oh. 

2 

3 

4 

5 

15" 

20 

26 

39 

107 

21" 

25 

'33 

47 

1'20 

27" 

32 

39 

56 

1’38 

33" 

38 

47 

107 

2 00 

40" 

46 

56 

1 '20 

2 34 

48" 

55 

1'07 

1 36 

3 29 

57" 

105 

1 21 

1 59 

5 14 

1'08" 

1 18 

1 38 

2 32 

10 16 

1 21" 

1 35 

2 00 

3 25 

For Latitude 37° 30'. 

Oh. 

2 

3 

4 

5 

18" 

22 

29 

43 

I'll 

24" 

28 

36 

51 

1'26 

30" 

35 

43 

101 

1 54 

36" 

42 

52 

113 

2 10 

44" 

50 

1'02 

1 27 

2 49 

52" 

100 

1 14 

1 49 

3 55 

102" 

112 

1 29 

2 14 

6 15 

114" 

1 26 

1 49 

2 54 

14 58 

1'29" 

1 45 

2 16 

4 05 

For Latitude 40°. 

Oh. 

2 

3 

4 

5 

21" 

25 

33 

47 

115 

27" 

’ 32 

40 

55 

1'31 

33" 

39 

48 

1 '06 

1 51 

40" 

46 

57 

1'19 

2 20 

48" 

52 

1'08 

1 36 

3 05 

57" 

1 06 

1 21 

1 58 

4 25 

108" 

1 19 

1 38 

2 30 

7 34 

1'21" 

1 35 

2 02 

3 21 

25 18 

1'39" 

1 57 
236 

4 59 

For Latitude 42° 30'. 

Oh. 

2 

3 

4 

5 

24" 

28 

36 

50 

116 

30" 

35 

43 

100 

1 36 

36" 

39 

52 

I'll 

1 58 

44” 

50 

ro2 

1 26 

2 30 

52" 

l'OO 

1 13 

1 44 

3 22 

102" 

1 12 

1 29 

2 10 
500 

114" 

1 26 

1 49 

2 49 

9 24 

1'29 ' 

1 45 

2 17 

3 55 

1’49" 

2 11 

2 59 

6 16 

For Latitude 45°. 

Oh. 

2 

3 

4 

5 

27" 

32 

40 

54 

1’23 

33" 

39 

47 

104 

1 41 

40" 

46 

56 

116 

2 05 

48" 

52 

107 

1 33 

2 41 

57" 

106 

1 21 

1 54 

3 40 

1'08" 

1 19 

1 38 

2 24 

5 40 

1’21" 

1 35 

2 00 

3 11 

12 02 

1'39” 

1 57 

2 34 

4 38 

2’02" 

2 29 

3 29 

8 15 

For Latitude 47° 30" 

Oh. 

2 

3 

4 

5 

30" 

35 

43 

56 

1'27 

36" 

42 

51 

1'09 

1 46 

44” 

50 

l'Ol 

1 23 

2 12 

52" 

l'OO 

1 13 

1 40 

2 52 

1'02” 

1 12 

1 28 

2 05 

4 01 

114” 
1 26 

1 47 

2 40 

6 30 

129” 

1 45 

2 15 

3 39 
16 19 

1'49" 
2 01 

2 56 

5 37 

2'18” 

2 51 

4 08 

11 18 














































































































TABLE XII. ACREAGE OF OPEN DRAINS, 


97 

Showing Number of Acres served by drains having bottom widths from 
1 ft. to 10 ft , with side slopes of l to 1, on the supposition of 1 inch rain¬ 
fall in 21 hours, one-half of which reaches the drain. 

Computed by B. F. Welles, C.E., Marshall, Mich. 


Fall in feet 
per 

Bottom Widths. 

1 ft. 

2 ft. 

3 ft. 




2 1't. 

3 ft 

2 ft. 

3 ft. 

2 ft, 

3 ft. 

1 mi. 

100 ft. 

8 rd. 

deep. 

deep. 

deep. 

deep. 

deep. 

deep. 

1.6 

.030 

.04 

407 

981 

594 

1311 

780 

1649 

2.0 

.038 

.05 

462 

1105 

665 

1473 

879 

1861 

2.4 

.045 

.06 

508 

1218 

732 

1622 

968 

2047 

2.8 

.053 

.07 

553 

1319 

797 

1762 

1053 

2217 

3.2 

.060 

.08 

592 

1416 

853 

1889 

1128 

2377 

3 6 

.070 

.09 

631 

1505 

939 

2009 

1198 

2529 

4.0 

.076 

.10 

666 

1590 

959 

2115 

1264 

2665 

4.8 

.091 

.12 

733 

1748 

1057 

2333 

1391 

2935 

5.6 

.110 

.14 

794 

1895 

1143 

2523 

1499 

3172 

6.4 

.120 

.16 

852 

2030 

1225 

2700 

1612 

3401 

7 2 

.136 

.18 

905 

2154 

1300 

2869 

1715 

3612 

8.0 

.150 

.20 

956 

2273 

1373 

3031 

1809 

3815 






Bottom Widths. 



TPoll in Vnni 

















per 


4 ft. 

5 

ft. 

6 ft. 




2 ft. 

3 ft. 

2 ft 

3 ft. 

2 ft. 

3 ft. 

l mi. 

100 ft. 

8 rd. 

deep. 

deep. 

deep. 

deep. 

deep. 

deep. 

1.6 

.030 

.04 

976 

2003 

1171 

2357 

1368 

2716 

2.0 

.038 

.05 

1094 

2249 

1316 

2650 

1541 

3046 

2.4 

.045 

.06 

1206 

2477 

1448 

2910 

1699 

3362 

2.8 

.053 

.07 

1308 

2684 

1572 

3158 

1835 

3642 

3.2 

.060 

.08 

1404 

2872 

1684 

3384 

1970 

3908 

3.0 

.070 

.09 

1494 

3049 

1790 

3598 

2097 

4150 

4.0 

.076 

.10 

1579 

3227 

1894 

3800 

2211 

4322 

4.8 

.091 

.12 

1731 

3553 

2089 

4173 

2436 

4810 

5.6 

.110 

.14 

1878 

3849 

2257 

4512 

2632 

5203 

6.4 

.120 

.16 

2013 

4115 

2415 

4838 

2820 

5571 

7 2 

.136 

.18 

2137 

4372 

2566 

5141 

3001 

5927 

8.0 

.150 

.20 

2256 

4609 

2705 

5412 

3165 

6257 






Bottom Widths. 



in 

















per 


7 ft. 

8 ft. 

10 ft. 




2 ft. 

3 ft. 

2 ft. 

3 ft. 

2 ft. 

3 ft. 

1 mi. 

100 ft. 

8 rd. 

deep. 

deep. 

deep. 

deep. 

deep. 

deep. 

1.0 

.030 

.04 

1574 

3074 

1767 

3458 

2177 

4179 

2.0 

.038 

.05 

1768 

3469 

1983 

3877 

2448 

4710 

2.4 

.045 

.06 

1946 

3807 

2181 

4265 

2695 

5169 

2.8 

.053 

.07 

2115 

4131 

2369 

4622 

2921 

5609 

3.2 

.060 

.08 

2259 

4427 

2538 

4948 

3136 

6014 

3.6 

.070 

.09 

2403 

4695 

2697 

5258 

3327 

6378 

4.0 

.076 

.10 

2538 

4963 

2848 

5552 

3508 

6745 

4.8 

.091 

.12 

2792 

5443 

3130 

6094 

3857 

7405 

5.6 

.110 

.14 

3029 

5894 

3393 

6591 

4184 

8010 

6.4 

.120 

.16 

3240 

6317 

3628 

7057 

4489 

8578 

7.2 

.136 

.18 

3443 

6715 

3854 

75C7 

4760 

9110 

8.0 

.150 

.20 

3629 

7078 

4070 

7910 

5038 

9623 


iafxomjYi Q-av. 

Foiu.ulas: v J -0.11. 4 — Q x 47.6 — Acreage. 
































































































98 TABLE XIII. ACREAGE OF TILE DRAINS, 


Showing Number of Acres drained by different sizes of tile, the rainfall 
being considered as equal Uj one-half inch in depth each 24 hours. 
Computed by R. C. Carpenter, Lansing, Mich. 


Rate of Inclination. 

Acres Drained. 





2-in 

3-ill. 

4-in. 

6-in. 

8-ill. 

10-ill. 

12-in. 

r eel to one oi rise. 

Tile. 

Tile. 

Tile. 

Tile. 

Tile. 

Tile. 

Tile. 

1 

font 

in to 

feet 

G.G 

18.9 






1 

4ft 

20 

ftft 

4.7 

13.0 

26.8 





1 

ftft 

25 

ftft 

4.2 

11.4 

24.0 

66.2 




1 

ftft 

30 

ftft 

39 

10.9 

21.9 

61.5 

126.4 



1 

ftft 

40 

ftft 

3.4 

9.4 

19.0 

53.3 

109.6 

190.5 


1 

ftft 

50 

ftft 

3.0 

8.4 

17.0 

47.7 

98.0 

170.4 

269.0 

1 

ftft 

GO 

• ft 

2.7 

7.G 

15.6 

43.4 

90.0 

156.0 

246.0 

1 

ftft 

70 

ftft 

2 5 

G.9 

14.5 

39.9 

83.0 

144.4 

228.1 

1 

ftft 

80 

ftft 

2.3 

6.5 

13.4 

37.2 

77.0 

135.0 

213.0 

1 

ft* 

90 

ftft 

2.2 

6.1 

12.6 

35.0 

72.5 

127.0 

200.5 

1 

ftft 

100 

1ft 

2.0 

5.7 

11.9 

33.1 

69.2 

120.6 

190.5 

1 

ftft 

150 

ftft 

1.6 

4.5 

9.5 

26.6 

56.0 

97.3 

154.4 

1 

ftft 

200 

• ft 


3.9 

8.2 

22.8 

48.0 

83.9 

132.5 

1 

ft* 

250 

ftft 


3.5 

7.5 

20.4 

43.4 

74.4 

117.0 

1 

•ft 

300 

ftft 



6.9 

18.4 

38 2 

65.5 

107.0 

1 

ftft 

400 

• ft 



5.9 

16.5 

34.6 

60 3 

90 7 

1 

ftft 

500 

ftft 




14.8 

30 1 

54 0 

81 6 

1 

ftft 

GOO 

ii 




13.3 

28.0 

48 6 

74.0 

1 

ftft 

800 

ftft 




24.0 

41.9 

65.0 

1 

ftft 

1,000 

ftft 





21.2 

37.2 

56.0 

1 

ftft 

1,500 

ftft 





30.8 

47 0 

1 

ftft 

2,000 

ftft 






40.8 









Note.— Tile should not be laid to grades where numbers are re 
placed by dashes. 


TABLE XIV. CArACITY OF TILE. 

Showing carrying capacity of different sizes of tile, in gallons. From 
Catalogue of the Bennett Sewer Pipe Co ., Jackson, Mich. 


Carrying Capacity—Gallons per Minute. 


Size of pipes. 

1 % in. fall per 
100 ft. 

I 

3 in. fall per 

100 ft. 

G in. fall per 

100 ft. 

9 in. fall per 

100 ft. 

1 ft. fall per 

100 ft. 

S-, 

o> 

Pa 

•*-< . 

^8 

VH 

2 ft. fall per 

100 ft. 

3 ft. fall per 

100 ft. 

2*4in. 

14 

20 

28 

34 

40 

49 

55 

68 

3 “ 

21 

30 

42 

52 

60 

74 

85 

104 

4 “ 

36 

52 

76 

92 

108 

132 

148 

184 

5 “ 

54 

78 

111 

134 

159 

192 

219 

269 

6 “ 

84 

120 

169 

206 

240 

294 

338 

414 

8 “ 

144 

208 

304 

368 

432 

528 

592 

736 

9 “ 

232 

330 

470 

570 

660 

810 

930 

1140 

10 “ 

267 

378 

463 

655 

803 

926 

1340 

1613 

12 “ 

470 

680 

960 

1160 

1360 

1670 

1920 

2350 

15 “ 

830 

1180 

1680 

2040 

2370 

2920 

3340 

4100 

18 “ 

1300 

1850 

2630 

3200 

3740 

4600 

5270 

6470 

20 “ 

1760 

2450 

3450 

4180 

4860 

5980 

6850 

8410 

24 “ 

3000 

4152 

5871 

7202 

8303 

10021 

11743 

14466 































































TABLE XV. AZIMUTHS OF TANGENT. 99 


Lati¬ 

tude. 

l mile. 

2 miles. 

3 miles. 

4 miles. 

o 

e 

• 

t 9 

O 9 

9 $ 

o 

9 

1 9 

o 

r 

9 9 

30 

89 

59 

30 

89 58 

59.9 

89 

58 

29.9 

89 

57 

59.9 

31 


59 

28.8 

58 

57.5 


58 

26.3 


57 

55.0 

32 


59 

27.5 

58 

55.0 


58 

22.5 


57 

50.0 

33 


59 

26.2 

58 

52.5 


58 

18.7 


57 

44.9 

34 


59 

24.9 

58 

49.9 


58 

14.8 


57 

39.7 

35 


59 

23.6 

58 

47.2 


58 

10.8 


57 

34.4 

36 


59 

22.2 

58 

44.4 


58 

06.8 


57 

28.9 

37 


59 

20.8 

58 

41.6 


58 

02.5 


57 

23.3 

38 


59 

19.4 

58 

38.8 


57 

58.2 


57 

17.5 

39 


59 

17.9 

58 

35.8 


57 

53.7 


57 

11.6 

40 


59 

16.4 

58 

32.8 


57 

49.2 


57 

05.5 

41 


59 

14 8 

58 

29.6 


57 

44.4 


56 

59.3 

42 


59 

13.2 

58 

26.4 


57 

39.6 


56 

52.8 

43 


59 

11.5 

58 

23.1 


57 

34.6 


56 

46.2 

44 


59 

09.8 

58 

19.6 


57 

29.5 


56 

39.3 

45 


59 

08.0 

58 

16.1 


57 

24.1 


56 

32.1 

46 


59 

06.2 

58 

12.4 


57 

18.6 


56 

24.8 

47 

89 

59 

04.3 

1 89 58 

08.6 

89 

57 

12.9 

89 

56 

17.1 

Lati¬ 

tude. 

5 miles. 

6 miles. 

7 miles. 

8 miles. 

o 

0 

9 

» » 

O 9 

,, 

o 

» 

9 9 

o 


9 9 

30 

89 

57 

29.9 

89 56 

59.8 

89 

56 

29.8 

89 

55 

59.8 

31 


57 

23.8 

56 

52.5 


56 

21.3 


55 

50.0 

32 


57 

17.5 

56 

45.0 


56 

12.5 


55 

40.0 

33 


57 

11.2 

56 

37.4 


56 

03.6 


55 

29.9 

34 


57 

04.6 

56 

29.6 


55 

54.5 


55 

19.4 

35 


56 

58.0 

56 

21.6 


55 

45.2 


55 

08.8 

36 


56 

51.1 

56 

13.4 


55 

35.6 


54 

57.8 

37 


56 

44.1 

56 

05.0 


55 

25.8 


54 

46.6 

38 


56 

36.9 

55 

56.3 


55 

15.7 


54 

35.1 

39 


56 

29.6 

55 

47.5 


55 

05.4 


54 

23 3 

40 


56 

21.9 

55 

38.3 


54 

54.7 


54 

11.1 

41 


56 

14.1 

55 

28.9 


54 

43.7 


53 

58.5 

42 


56 

06.0 

55 

19.2 


54 

32.4 


53 

45.6 

43 


55 

57.7 

55 

09.2 


54 

20.8 


53 

32.3 

44 


55 

49.1 

54 

58.9 


54 

08.7 


53 

18.5 

45 


55 

40.2 

54 

48.2 


53 

56.3 


53 

04.3 

46 


55 

31.0 

54 

37.2 


53 

43.4 


52 

49.5 

47 

89 

55 

21.4 

89 54 

25.7 

89 

53 

30.0 

89 

52 

34.3 

Lati¬ 

tude. 

9 miles. 

10 miles. 

11 miles. 

12 miles. 

• 

• 

9 

9 9 

o » 

9 1 

o 


9 9 

o 

1 

9 9 

30 

89 

55 

29.8 

89 54 

59.7 

89 

54 

29.7 

89 

53 

59.7 

31 


55 

18.8 

54 

47.6 


54 

16.3 


53 

45.1 

32 


55 

07.6 

54 

35.1 


54 

02.6 


53 

30.1 

33 


54 

56.1 

54 

22.3 


53 

48.5 


53 

14.8 

34 


54 

44.4 

54 

09.3 


53 

34.2 


52 

59.1 

35 


54 

32.3 

53 

55.9 


53 

19.5 


52 

43.1 

36 


54 

20.0 

53 

42.3 


53 

04.5 


52 

26.7 

37 


54 

07.4 

53 

28.2 


52 

49.1 


52 

09.9 

38 


53 

54.5 

53 

13.9 


52 

33.2 


51 

52.6 

39 


53 

41.2 

52 

59.1 


52 

17.0 


51 

34.9 

40 


53 

27.5 

52 

43.8 


52 

00.2 


51 

16.6 

41 


53 

13.4 

52 

28.2 


51 

43.0 


50 

57.8 

42 


52 

58.8 

52 

12.0 


51 

25.2 


50 

38.4 

43 


52 

43.8 

51 

55.4 


51 

06.9 


50 

18.5 

44 


52 

28.4 

51 

38.2 


50 

48.0 


49 

57.8 

45 


52 

12.3 

51 

20.4 


.50 

28.4 


49 

36.4 

46 


51 

55.7 

51 

01.9 


50 

08.1 


49 

14.3 

47 

89 

51 

38.6 

89 50 

42.9 

89 

49 

47.2 

89 

48 

51.4 































































100 TABLE XVI. OFFSETS FliOM TANGENT. 


Lati¬ 

tude. 

1 mile. 

2 miles. 

3 miles. 

4,miles. 

• 

Feet. 

Feet. 

Feet. 

Feet. 

30 

0.39 

1.54 

3.47 

6.17 

31 

0.40 

1.00 

3.61 

6.42 

32 

0.42 

1.67 

3.76 

6.67 

33 

0.43 

1.73 

3.90 

6.93 

34 

0.45 

1.80 

4.05 

7.20 

35 

0.47 

1.87 

4.20 

7.47 

30 

0.48 

1.94 

4.36 

7.75 

37 

0.50 

2.01 

4.52 

8.04 

38 

0.52 

2.08 

4.69 

8.33 

39 

0.54 

2.10 

4.86 

8.63 

40 

0.50 

2.24 

5.03 

8.95 

41 

0.58 

2.32 

5.21 

9.27 

42 

0.00 

2.40 

5.40 

9.59 

43 

0.02 

2.48 

5.59 

9.93 

44 

0.04 

2.57 

5.79 

10.29 

45 

0.07 

2.60 

5.99 

10.05 

40 

0.09 

2.76 

6.20 

11.02 

47 

0.71 

2.85 

6.42 

11.41 

Lati¬ 

tude. 

5 miles. 

6 miles. 

7 miles. 

8 miles. 

o 

Feet. 

Feet. 

Feet. 

Feet. 

30 

9.04 

13.88 

18.89 

24.67 

31 

10.03 

14.44 

19.60 

25.68 

32 

10.42 

15.02 

20.44 

26.69 

33 

10.82 

15.00 

21.23 

27.74 

34 

11.25 

16.20 

22.06 

28.80 

35 

11.08 

10.81 

22.89 

29.89 

30 

12 11 

17.41 

23.74 

31.01 

37 

12.57 

18.09 

24.62 

32.16 

38 

13.02 

18.75 

25.52 

33.33 

39 

13.49 

19.43 

26.44 

34.54 

40 

13.98 

20.11 

27.40 

35.78 

41 

14.48 

20.85 

28.37 

37.06 

42 

14.99 

21.59 

29.38 

38.38 

43 

15.52 

22.35 

30.42 

39.74 

44 

10.07 

23.14 

31.50 

41.14 

45 

16.04 

23.90 

32.61 

42.59 

40 

17.21 

24.80 

33.7S 

44.10 

47 

17.83 

25.68 

34.95 

45.65 

Lati¬ 

tude. 

9 miles. 

10 miles. 

11 miles. 

12 miles. 

o 

Feet. 

Feet. 

Feet. 

Feet. 

30 

31.23 

38.55 

40.05 

55.52 

31 

32.49 

40.12 

48.54 

57.77 

32 

33.78 

41.71 

50.47 

60.06 

33 

35.10 

43.34 

52.44 

62.41 

34 

30.45 

45.00 

54.45 

64.80 

35 

37.83 

40.71 

50.02 

67.26 

30 

39.25 

48.45 

58.63 

69.77 

37 

40.70 

50.24 

60.79 

72.35 

38 

42.19 

52.08 

63.02 

75.00 

39 

43.71 

53.97 

65.30 

77.71 

40 

45.29 

55.91 

67.65 

80.51 

41 

40.90 

57.91 

70.07 

83.39 

42 

48.57 

59.97 

72.50 

86.35 

43 

50.29 

62.09 

75.13 

89.41 

44 

62.07 

64.28 

77.78 

92.57 

45 

53.91 

66.55 

80.53 

95.84 

40 

55.81 

68.90 

83.37 

99.22 

47 

57.78 

71.34 

86.32 

102.72 







































Table xvii.— Minutes in Decimals of a Degree. TO 1 


1' 

.0107 


ir 

.1833 


21' 

.3500 

31' 

,5167 


41' 

.6833 


51' 

.8500 

2 

.0333 


12 

.2000 


22 

.3667 

32 

.5333 


42 

.7000 


52 

.8667 

:l 

.0500 


13 

.2167 


23 

.3833 

33 

.5500 


43 

.7167 


53 

.8833 

4 

.0067 


14 

.2333 


24 

.4000 

34 

.5667 


44 

.7333 


54 

.9000 

5 

.0833 


15 

.2500 


25 

.4167 

35 

.5833 


45 

.7500 


55 

.9167 

G 

.1000 


1G 

.2667 


26 

.4333 

36 

.6000 


4<> 

.7667 


5G 

.9333 

7 

.1167 


17 

.2833 


27 

.4500 

37 

.6167 


47 

.7833 


57 

.9500 

8 

.1333 


18 

.3000 


28 

.4667 

38 

.6333 


48 

,8000 


58 

,9667 

9 

.1500 


19 

.3167 


29 

.4833 

39 

.6500 


49 

.8167 


59 

.9833 

10 

.1667 


20 

.3333 


30 

.5000 

I 40 

.6667 


50 

.8333 


60 

1.0000 


Table xviii. — Inches in Decimals of a Foot. 

1-16 

.0052 

3-32 

.0078 

.01^4 

3-16 

.0156 

.0208 

5-16 

.0260 

% 

.0313 

A 

0417 

% 

.0521 

.o4s 

% 

.0729 

l 

.0833 

2 

.1667 

3 

.2500 

4 

.3333 

5 

.4167 

6 

.5000 

7 

.5833 

8 

.6667 

9 

.7500 

10 

.8333 

11 

.9167 

Table xix.— Radii, and Deflections. 


Deg 

Radius 

Tan. 

Def 

Chd. 

Def. 

Def. 

for 

1 Foot 

Deg. 

Radius 

Tan. 

Def. 

Chd. 

Def. 

Def. 

for 

l Foot 

0° 10' 

34377. 

.145 

.291 

0.05' 

7° 

819.0 

6.105 

12.21 

2.10' 

20 

17189. 

.291 

.582 

0.10 

20' 

781.8 

6.395 

12.79 

2.20 

30 

11459. 

.436 

.873 

0.15 

30 

764.5 

6.540 

13.08 

2.25 

40 

8594.4 

.582 

1.164 

0.20 

40 

747 9 

6.685 

13.37 

2.30 

50 

6875.5 

.727 

1.454 

0.25 

8 

716.8 

6.976 

13.95 

2.40 

1 

5729 6 

.873 

1 745 

0.30 

20 

688.2 

7.266 

14.53 

2.50 

10 

4911.2 

1.018 

2.036 

0.35 

30 

674.7 

7.411 

14.82 

2.55 

20 

4297.3 

1.164 

2.327 

0.40 

40 

661.7 

7.556 

15.11 

2 60 

30 

3819.8 

1.309 

2.618 

0.45 

9 

637.3 

7.846 

15.69 

2.70 

40 

3437.9 

1.454 

2.909 

0.50 

20 

614.6 

8.136 

16,27 

2.80 

50 

3125.4 

1.600 

3.200 

0.55 

30 

603.8 

8.281 

16.56 

2.85 

2 

2864.9 

1.745 

3.490 

0.60 

40 

593.4 

8.426 

16.85 

2.90 

10 

2644.6 

1.891 

3.781 

0.65 

10 

573 7 

8.716 

17.43 

3.00 

20 

2455.7 

2.036 

4 072 

0.70 

30 

546.4 

9.150 

18.30 

3.15 

30 

2292.0 

2.181 

4.363 

0.75 

11 

521.7 

9.585 

19.16 

3.30 

40 

2148.8 

2.327 

4.654 

0.80 

30 

499.1 

10.02 

20.04 

3.45 

50 

2022.4 

2.472 

4 945 

0.85 

12 

478.3 

10.45 

20.91 

3.60 

3 

1910 1 

2.618 

5.235 

0.90 

30 

459.3 

10.89 

21.77 

3.75 

10 

1809.6 

2.763 

5.526 

0.95 

13 

441.7 

11.32 

22.64 

3.90 

20 

1719.1 

2.908 

5.817 

1.00 

30 

425.4 

11.75 

23.51 

4.05 

39 

1637.3 

3.054 

6.108 

1.05 

14 

410.3 

12.18 

24.37 

4.20 

40 

1562.9 

3.199 

6.398 

1.10 

30 

396.2 

12.62 

25.24 

4.35 

50 

1495.0 

3.345 

6.689 

1.15 

15 

383.1 

13.05 

26.11 

4.50 

4 

1432.7 

3 490 

6.980 

1.20 

30 

370.8 

13.49 

26.97 

4.65 

10 

1375.4 

3.635 

7.271 

1.25 

1G 

359.3 

13.92 

27.84 

4.80 

20 

1322.5 

3.781 

7.561 

1.30 

30 

348.5 

14.35 

28.70 

4.95 

30 

1273.6 

3.926 

7.852 

1,35 

17 

338.3 

14.78 

29.56 

5.10 

40 

1228.1 

4.071 

8.143 

1.40 

18 

319.6 

15.64 

31.29 

5.40 

50 

1185.8 

4.217 

8.433 

1.45 

19 

302.9 

16.51 

33.01 

5.70 

5 

1146.3 

4.362 

8.724 

1.50 

20 

287.9 

17.37 

34.73 

6.00 

10 

1109.3 

4.507 

9.014 

1.55 

21 

274.4 

18.22 

36.44 

6.30 

20 

1074.7 

4.653 

9.305 

1.60 

22 

262.0 

19.08 

38.16 

6.60 

30 

1042.1 

4.798 

9.596 

1.65 

23 

250.8 

19.94 

39.87 

6.90 

40 

1011.5 

4 943 

9.886 

1.70 

24 

240.5 

20.79 

41.58 

7.20 

50 

982.6 

5.088 

10.18 

1.75 

25 

231.0 

21.64 

43.28 

7.50 

6 

955.4 

5.234 

10.47 

1.80 

26 

222.3 

22.50 

44.99 

7.80 

10 

929.6 

5.379 

10.76 

1.85 

27 

214.2 

23.35 

46.69 

8.10 

20 

905.1 

5.524 

11.05 

1.90 

28 

206.7 

24.19 

48.38 

8.40 

30 

881.9 

5.669 

11.34 

1.95 

29 

199.7 

25.04 

50.07 

8.70 

40 

859.9 

5.814 

11.63 

2.00 

SO 

193.2 

25.88 

51.76 

9.00 


32 





















































































I 


102 Table xx —Tangents and Externals to a 1° Curve. 


Angle. 

Tangent. 

Eiter’l. 


Angle. 

Tangent. 

Eiternal 

Angle. 

Tangent, 

External 

1° 

50.00 

.22 


11° 

.551.70 

26.50 


‘21° 

1061.9 

97.57 

10' 

58.34 

.30 


10' 

560.11 

27.31 


10' 

1070.6 

99.16 

20 

60.67 

.39 


20 

568.53 

28.14 


20 

1079 2 

100.75 

30 

75.01 

.49 


30 

576.95 

28 97 


30 

1087.8 

102.35 

40 

83.34 

.61 


40 

.585.36 

29.82 


40 

1096.4 

103 97 

50 

91.68 

.73 


50 

593.79 

30.68 


50 

1105.1 

105.00 

2 

100.01 

.87 


12 

602.21 

31.56 


22 

1113.7 

107 24 

10 

108.35 

1.02 


10 

610.64 

32.45 


10 

1122.4 

108 90 

20 

116.68 

1.19 


20 

619.07 

33.35 


20 

1131.0 

110.57 

30 

125.02 

1.36 


30 

627.50 

34.26 


30 

1139.7 

112 25 

40 

133.36 

1.55 


40 

635.93 

35.18 


40 

1148.4 

113 95 

50 

141.70 

1.75 


50 

614.37 

36.12 


50 

1157.0 

115.66 

3 

150.04 

1.96 


13 

652.81 

37 07 


23 

1165.7 

117.38 

10 

158.38 

2.19 


10 

661.25 

38.03 


10 

1174.4 

119.12 

20 

166.72 

2.43 


20 

669.70 

39.01 


20 

1183.1 

120 87 

30 

175.06 

2.67 


30 

678.15 

39.99 


30 

1191.8 

122 63 

40 

183.40 

2.93 


40 

686 60 

40.99 


40 

1200.5 

124 41 

50 

191.74 

3.21 


50 

695.06 

42.00 


50 

1209,2 

126.20 

4 

200.08 

3.49 


14 

703 51 

43.03 


24 

1217.9 

128 00 

10 

208.43 

3.79 


10 

711.97 

44.07 


10 

1226.6 

129.82 

20 

216.77 

4.10 


20 

720.44 

45 12 


20 

1235.3 

131.65 

30 

225.12 

4.42 


30 

728.90 

46.18 


30 

1244.0 

133.50 

40 

233.47 

4.76 


40 

737.37 

47.25 


40 

1252.8 

135.35 

50 

241 81 

5.10 


50 

745.85 

48.34 


50 

1261.5 

137.23 

5 

250.16 

5.46 


15 

754.32 

49.44 


25 

1270.2 

139 11 

10 

258.51 

5.83 


10 

762.80 

50.55 


10 

1279.0 

141.01 

20 

266.86 

6.21 


20 

771.29 

51.68 


20 

1287 7 

142.93 

30 

275.21 

6.61 


30 

779.77 

52.89 


30 

1296.5 

144.85 

40 

283.57 

7.01 


40 

788.26 

53.97 


40 

1305.3 

146.79 

50 

291.92 

7.43 


50 

796.75 

55.13 


50 

1314.0 

148.75 

6 

300.28 

7.86 


16 

805.25 

56 31 


26 

1322.8 

150 71 

10 

308.64 

8.31 


10 

813.75 

57.50 


10 

1331.6 

152.69 

20 

316.99 

8.76 


20 

822.25 

58.70 


20 

1340.4 

154.69 

30 

325.35 

9.23 


30 

830.76 

59.91 


30 

1349.2 

156.70 

40 

333.71 

9 71 


40 

839.27 

61.14 


40 

1358.0 

158.72 

50 

342.08 

10.20 


50 

847.78 

62.38 


50 

1366.8 

160.76 

7 

350.44 

10.71 


17 

856.30 

63.63 


27 

1375.6 

162.81 

10 

3.58.81 

11.22 


10 

864.82 

64.90 


10 

1384.4 

164.86 

20 

367.17 

11.75 


20 

873.35 

66 18 


20 

1393.2 

166.95 

30 

375.54 

12.29 


30 

881.88 

67.47 


30 

1402.0 

169.04 

40 

383.91 

12.85 


40 

890.41 

68.77 


40 

1410.9 

171.15 

. 50 

392.28 

13.41 


50 

898.95 

70.09 


50 

1419.7 

173.27 

8 

400.66 

13.99 


18 

907.49 

71.42 


28 

1428.6 

175.41 

10 

409.03 

14.58 


10 

916.03 

72.76 


10 

1437.4 

177.55 

20 

417.41 

15.18 


20 

924.58 

74.12 


20 

1446 3 

179.72 

30 

425.79 

15.80 


30 

933.13 

75.49 


30 

1455.1 

181.89 

40 

434.17 

16.43 


40 

941.69 

76.86 


40 

1464 0 

184.08 

50 

442.55 

17.07 


50 

950.25 

78.26 


50 

1472.9 

186 29 

9 

450.93 

17.72 


19 

958.81 

79.67 


29 

1481.8 

188.51 

10 

459 32 

18.38 


10 

967,38 

81.09 


10 

1490.7 

190.74 

20 

467.71 

19.06 


20 

975.96 

82-53 


20 

1499.6 

192.99 

30 

476.10 

19.75 


30 

984.53 

83.97 


30 

1508.5 

195.25 

40 

484.49 

20.45 


40 

993.12 

85.43 


40 

1517.4 

197.53 

50 

492.88 

21.16 


50 

1001.7 

86.90 


50 

1526.3 

199.82 

10 

501.28 

21.89 


20 

1010.3 

88.39 


30 

1535.3 

202 12 

10 

509.68 

22.62 


10 

1018.9 

89.89 


10 

1544.2 

204.44 

20 

518.08 

23.38 


20 

1027.5 

91.40 


20 

1553.1 

206.77 

30 

526.48 

24.14 


30 

1036.1 

92.92 


30 

1562.1 

209.12 

40 

534.89 

24.91 


40 

1044.7 

94.46 


40 

1571.0 

211.48 

50 

513.29 

25.70 


50 

10.53.3 

96 01 


50 

1580.0 

213.86 


































Table xx—Tangents and Externals to a 1° Curve. 103 


l ingle. 

Tangent. Exter’l. 

Angle. 

Tangent. 

External 

Angle. 

| Tangent, 

External 

31° 

1589.0 

216 3 

41° 

2112.2 

387.4 

51° 

2732.9 

618.4 

10' 

1598.0 

218.7 

10' 

2151.7 

390.7 

10' 

2743.1 

622.8 

20 

1606.9 

221.1 

20 

2161,2 

394.1 

20 

2753.4 

6:7.2 

30 

1615 9 

223.5 

30 

2170.8 

397.4 

30 

2763.7 

631.7 

40 

1624 9 

226.0 

40 

2180.3 

400.8 

40 

2773.9 

636.2 

50 

1633.9 

228.4 

50 

2189.9 

401.2 

50 

2784.2 

640.7 

32 

1643.0 

230.9 

42 

2199.4 

407.6 

52 

2794.5 

615.2 

10 

1652.0 

233.4 

10 

2209 0 

411.1 

10 

2804.9 

649.7 

20 

1661.0 

235.9 

20 

2218.6 

411.5 

20 

2815.2 

614.3 

30 

1670.0 

238.4 

30 

2228.1 

418.0 

30 

2825.6 

658 8 

40 

1679.1 

211.0 

40 

2237.7 

421.4 

10 

2835.9 

663.4 

50 

1688.1 

243.5 

50 

2247.3 

425.0 

50 

2846.3 

668.0 

33 

1697.2 

246 1 

43 

2257.0 

428.5 

53 

2856.7 

672 7 

10 

1706.3 

248.7 

10 

2266.6 

432.0 

10 

2867.1 

677.3 

20 

1715 3 

251.3 

20 

2276.2 

435 6 

20 

2877.5 

682.0 

30 

1724.4 

253.9 

30 

2235.9 

439.2 

30 

2888.0 

686.7 

40 

1733 5 

256.5 

40 

2295.6 

442 8 

40 

2898.4 

691.4 

50 

1742.6 

259.1 

50 

2305.2 

446.4 

50 

2908 9 

696.1 

34 

1751.7 

261.8 

44 

2314.9 

450.0 

54 

2919.4 

700.9 

10 

1760.8 

264.5 

10 

2324.6 

453.6 

10 

2929 9 

705.7 

-.0 

1770 0 

267.2 

20 

2334.3 

457.3 

20 

2910.4 

710.5 

30 

1779.1 

269.9 

30 

2344.1 

461.0 

30 

2951.0 

715.3 

40 

1788 2 

272.6 

40 

2353 8 

464.6 

40 

2961.5 

720.1 

50 

1797.4 

275.3 

50 

2363.5 

468.4 

50 

2972.1 

725.0 

35 

1806 6 

278.1 

45 

2373.3 

472.1 

55 

2982.7 

729.9 

10 

1815.7 

280.8 

10 

2383.1 

475.8 

10 

2903.3 

734.8 

20 

1824.9 

283.6 

20 

2392.8 

479.6 

20 

3003.9 

739.7 

30 

1834.1 

286.4 

30 

2102.6 

483.8 

30 

3014.5 

744.6 

40 

1843.3 

289.2 

40 

2112.4 

487.2 

40 

3025.2 

719.6 

1 50 

1852.5 

292.0 

50 

2422.3 

491,0 

50 

3035.8 

754.6 

36 

1861.7 

294 9 

46 

2432.1 

494.8 

56 

3016.5 

759 6 

10 

1870.9 

297.7 

10 

2411.9 

498.7 

10 

3057.2 

764 6 

20 

1880.1 

300.6 

20 

2451.8 

502.5 

20 

3067,9 

769.7 

30 

1889.4 

303.5 

30 

2461.7 

506.4 

30 

3078.7 

774.7 

40 

1898.6 

306.4 

40 

2471.5 

510.3 

40 

3089.4 

779.8 

50 

1907.9 

309.3 

50 

2481.4 

514.3 

50 

3100.2 

781.9 

37 

1917.1 

312.2 

47 

2491.3 

518.2 

57 

3110.9 

790.1 

10 

1926.4 

315.2 

10 

2501.2 

522.2 

10 

3121 7 

795.2 

20 

1935.7 

318.1 

20 

2511.2 

526.1 

20 

3132.6 

800.4 

30 

1945.0 

321.1 

30 

2521.1 

530.1 

30 

3143.4 

805.6 

40 

9454 3 

324 1 

40 

2531.1 

531.2 

40 

3154.2 

810.9 

50 

1963.6 

327.1 

50 

2541.0 

538.2 

50 

3165.1 

816.1 

38 

1972.9 

330.2 

48 

2551.0 

512.2 

58 

3176.0 

821.4 

10 

1982.2 

333.2 

10 

2.561 0 

516.3 

10 

3186.9 

826.7 

20 

1991.5 

336.3 

20 

2571.0 

550 4 

20 

3197.8 

832 0 

30 

2000.9 

339.3 

30 

2581.0 

554.5 

30 

3208 8 

837 3 

40 

2010.2 

342.4 

40 

2591.0 

558.6 

40 

3219.7 

812.7 

50 

2019.6 

345.5 

50 

2601.1 

562.8 

50 

3230.7 

848.1 

30 

2029.0 

318.6 

49 

2611.2 

566.9 

5!) 

3241.7 

853.5 

10 

2038.4 

351.8 

10 

2621.2 

571.1 

10 

3252 7 

8.58.9 

20 

2047.8 

354 9 

20 

26 51.3 

575.3 

20 

3263.7 

861.3 

30 

2057.2 

358 l 

30 

2611.4 

579.5 

30 

3274.8 

869.8 

40 

2006 6 

361.3 

40 

2651 5 

583.8 

40 

3285.8 

875.3 

50 

2076.0 

361.5 

50 

2661.6 

588.0 

50 

3296,9 

880.8 

40 

2085.4 

367.7 

50 

2671.8 

592.3 

60 

3308.0 

886.4 

10 

2091.9 

371.0 

10 

2681.9 

596.6 

10 

3319 1 

892.0 

20 

2101 3 

374 2 

20 

2692.1 

600.9 

20 

3:330.3 

897.5 

30 

2113.8 

377.5 

30 

2702.3 

605 3 

30 

3341.4 

903.2 

40 

2123.3 

380.8 

40 

2712 5 

609 6 

40 

3352 6 

908.8 

50 

2132.7 

384.1 

50 

2722.7 

614.0 

50 

3363.8 

914,5 

1 



















































104 Table xx— Tangents and Externals to a 1° Curve 


Angle. 

Tangent, 

External 

Angle. 

Tangent. 

External 

Angle. 

Tangent. 

External 

61° 

3375.0 

920.2 

71° 

4086.9 

1308.2 

SV 

4893.6 

1805.3 

10' 

3386.3 

925.9 

10' 

4099.5 

1315 6 

10' 

4908.0 

1814.7 

20 

3397.5 

931 6 

20 

4112.1 

1322.9 

20 

4922.5 

1821.1 

30 

3408.8 

937.3 

30 

4124.8 

1330.3 

30 

4937.0 

1833 6 

40 

3420.1 

943.1 

40 

4137.4 

1337.7 

40 

4951.5 

1843.1 

50 

3431.4 

948.9 

50 

4150.1 

1345.1 

50 

4966.1 

1852.6 

62 

3442.7 

954.8 

72 

4162.8 

1352.6 

82 

4980.7 

1862.2 

10 

3454.1 

960.6 

10 

4175.6 

1360.1 

10 

4995.4 

1871.8 

20 

3465.4 

966.5 

20 

4188.5 

1367.6 

20 

5010 0 

1881.5 

30 

3476 8 

972.4 

30 

4201.2 

1375 2 

. 30 

5024.8 

1891.2 

40 

3488 3 

978.3 

40 

4214 0 

1382.8 

40 

5039.5 

1900.9 

50 

3499.7 

984 3 

50 

4226.8 

1390.4 

50 

5054 3 

1910,7 

63 

3511.1 

990.2 

73 

4239.7 

1398.0 

83 

5069.2 

1920.5 

10 

3522.6 

996.2 

10 

4252.6 

1405.7 

10 

5084.0 

1930.4 

20 

3 >34.1 

1002 3 

20 

4265-6 

1413.5 

20 

5u99.0 

1940.3 

30 

3545.6 

1008.3 

30 

4278.5 

1421.2 

30 

5113.9 

1950.3 

40 

3557.2 

1014.4 

40 

4291.5 

1429.0 

40 

5128.9 

1960.2 

50 

3568.7 

1020.5 

59 

4304.6 

1436.8 

50 

5143.9 

1970.3 

64 

&580.3 

1026.6 

74 

4317.6 

1414.6 

84 

5159.0 

1980.4 

10 

3591.9 

1032.8 

10 

4330.7 

1452.5 

10 

5174 1 

199J.5 

20 

3603.5 

1039 0 

20 

4343.8 

1460 4 

20 

5189.3 

2000.6 

30 

3615 1 

1045.2 

30 

4356.9 

1468.4 

30 

5204.4 

2010.8 

40 

3626.8 

1051.4 

40 

4370.1 

1476.4 

40 

5219.7 

2021.1 

50 

3638.5 

1057.7 

50 

4383.3 

1484.4 

50 

5234.9 

2031.4 

65 

3650.2 

1063.9 

75 

4396 5 

1492.4 

85 

5250.3 

2041.7 

10 

3661.9 

1070.2 

10 

4409.8 

1500.5 

10 

5265 6 

2052.1 

20 

3673.7 

1076.6 

20 

4423.1 

1508 6 

20 

5281.0 

2062.5 

30 

3685.4 

1082.9 

30 

4436 4 

1516.7 

30 

5296.4 

2073.0 

40 

3697.2 

1089.3 

40 

4449.7 

1524.9 

40 

5311 9 

2083.5 

50 

3709.0 

1095.7 

50 

4463.1 

1533.1 

50 

5327.4 

2094.1 

66 

3720.9 

1102.2 

76 

4476.5 

1541.4 

86 

5343.0 

2104.7 

10 

3732.7 

1108.6 

10 

4489.9 

1519.7 

10 

5358.6 

2115.3 

20 

3744.6 

1115.1 

20 

4503.4 

1558.0 

20 

5374.2 

2126 0 

30 

3756.5 

1121.7 

30 

4516.9 

1566.3 

30 

5689.9 

2136.7 

40 

3768.5 

1128.2 

40 

4530.4 

1574.7 

40 

5405.6 

2147.5 

50 

3780.4 

1134.8 

50 

4544.0 

1583.1 

50 

5121.4 

2158.4 

67 

3792.4 

1141.4 

77 

4557.6 

1591 6 

87 

5437.2 

2169.2 

10 

3804.4 

1148 0 

10 

4571.2 

1600.1 

10 

5453.1 

2180.2 

20 

3816 4 

1154.7 

20 

4584.3 

1608 6 

20 

5469.0 

2191.1 

30 

3828.4 

1161.3 

30 

4598.5 

1617.1 

30 

5481.9 

2202.2 

40 

3840.5 

1168.1 

40 

4612.2 

1625.7 

40 

5500 9 

2213.2 

50 

3852.6 

1174.8 

50 

4626.0 

1634.4 

50 

5517.0 

2224.3 

68 

3864.7 

4181.6 

78 

4639.8 

1643.0 

88 

5533.1 

2235.5 

10 

3876.8 

1188.4 

10 

4653.6 

1651.7 

10 

5549.2 

2246.7 

20 

3889.0 

1195.2 

20 

4667.4 

1660.5 

20 

5565.4 

2258.0 

30 

3901.2 

1202.0 

30 

4681.3 

1669.2 

30 

5581.6 

2269.3 

40 

3913.4 

1208.9 

40 

4695.2 

1678.1 

40 

5597.8 

2280.6 

50 

3925.6 

1215.8 

50 

4709.2 

1686.9 

50 

5614.2 

2292.0 

69 

3937.9 

1222.7 

79 

4723.2 

1695.8 

89 

5630.5 

2303.5 

10 

3950.2 

1229.7 

10 

4737.2 

1704.7 

10 

5646.9 

2315.0 

20 

3962.5 

1236.7 

20 

4751.2 

1713.7 

20 

5663.4 

2326.6 

30 

3974.8 

1243.7 

30 

4765.3 

1722.7 

30 

5679.9 

2338.2 

40 

3987.2 

1250.8 

40 

4779.4 

1731.7 

40 

5696 4 

2349.8 

50 

3999.5 

1257.9 

50 

4793.6 

1740.8 

50 

5713.0 

2361.5 

70 

4011.9 

1265.0 

80 

4807.7 

1749.9 

90 

5729.7 

2-373.3 

10 

4024.4 

1272.1 

10 

4822.0 

1759.0 

10 

5746.3 

2385.1 

20 

4036.8 

1279.3 

20 

4836.2 

1768.2 

20 

5763.1 

2397.0 

30 

4049.3 

1286.5 

30 

4850.5 

1777.4 

30 

5779.9 

2408.9 

40 

4061.8 

1293.6 

40 

4864.8 

1786.7 

40 

5796.7 

2420.9 

50 

4074.4 

1300.9 

50 

4879.2 

1796.0 

59 

5813.6 

2432.9 













































Table xx— Tangents and Externals to a 1° Curve. 105 


Angle. 

Tangent 

External 


Angle. 

Tangent 

External 


Angle. 

Tangen; 

External 

91° 

5830.5 

2444.9 


101° 

6950.6 

3278.1 


111° 

8336 7 

4386.1 

10' 

5847.5 

2457.1 


10' 

6971.3 

3294,1 


10' 

8362.7 

4407,6 

20 

5864.6 

2469.3 


20 

6992 0 

3310.1 


20 

8388.9 

4429.2 

30 

5881.7 

2481.5 


30 

7012.7 

3326.1 


30 

8115.1 

4450 9 

40 

5898.8 

2493.8 


40 

7033.6 

3342.3 


40 

8441.5 

4472.7 

50 

5916.0 

2506.1 


50 

7054.5 

3358.5 


50 

8465.0 

4494.6 

92 

5933.2 

2518.5 


102 

7075.5 

3374.9 


11*2 

8494.6 

4516 6 

10 

5950.5 

2-531.0 


10 

7096 6 

3391.2 


10 

8521.3 

4538.8 

20 

5967.9 

2543.5 


20 

7117.8 

3407.7 


20 

8548.1 

4561,1 

30 

5985.3 

2556.0 


30 

7139.0 

3424.3 


30 

8575.0 

4533.4 

40 

6002.7 

2568 6 


40 

7160 3 

3440.9 


40 

8602.1 

4606 0 

50 

6020.2 

2581.3 


50 

7181.7 

3457.6 


50 

8629 3 

4628.6 

93 

6037.S 

2594.0 


103 

7203.2 

3474.4 


113 

8656.6 

4651.3 

10 

6055.4 

2606 8 


10 

7224.7 

3491.3 


10 

8684.0 

4674.2 

20 

6073.1 

2619.7 


20 

7246.3 

3508.2 


20 

8711.5 

4697,2 

30 

6090.8 

2632 6 


30 

7288.0 

3525.2 


30 

8739.2 

4720.3 

40 

6108.6 

2645 5 


40 

7289 8 

3542,4 


40 

8767.0 

4743.6 

50 

6126.3 

2658.5 


50 

7311.7 

3559.6 


50 

8794.9 

4766-9 

94 

6144.3 

2671.6 


104 

7333 6 

3576.8 


114 

8822.9 

4790.4 

10 

6162.6 

2684.7 


10 

73)5.6 

3594.2 


10 

8851.0 

4814.1 

20 

6180.2 

2697.9 


20 

7377.8 

3611.7 


20 

8879 3 

4837.8 

30 

6198.3 

2711.2 


30 

7399.9 

3629.2 


30 

8907.7 

4861.7 

40 

6216.4 

2724.5 


40 

7422.2 

3646.8 


40 

8936.3 

4885.7 

50 

6234.6 

2737.9 


50 

7444.6 

3664.5 


50 

8965,0 

4909.9 

95 

6252.8 

2751.3 


105 

7467.0 

3682,3 


115 

8993.8 

4934.1 

10 

6271.! 

2764.8 


10 

7489.6 

3700.2 


10 

9022.7 

4958.6 

20 

6289.4 

2778.3 


20 

7512.2 

3718.2 


20 

9051.7 

4983.1 

SO 

6307.9 

2792 0 


30 

7534 9 

3736.2 


30 

9080.9 

5007.8 

40 

6326.3 

2865.6 


40 

7557.7 

3754.4 


40 

9110.3 

5032.6 

50 

6344.8 

2819.4 


50 

75^0.5 

3772,6 


50 

9139.8 

5057.6 

96 

6363.4 

2833,2 


106 

7603 5 

3791.0 


116 

9169.4 

5082.7 

10 

6382.1 

2847.0 


10 

7626.6 

3809.4 


10 

9199.1 

5107.9 

20 

6400 8 

2861,0 


20 

7649.7 

3827.9 


20 

9229.0 

5133,3 

30 

6419.5 

2875,0 


30 

7672,9 

3846.5 


30 

9259 0 

5158.8 

40 

6438.4 

2889,0 


40 

7696 3 

3865,2 


40 

9289.2 

5184.5 

50 

6457.3 

2903.1 


50 

7719.7 

3884,0 


50 

9319.5 

5210.3 

97 

64/6.2 

2917.3 


107 

7743,2 

3902.9 


117 

9349 9 

5236.2 

1) 

6495.2 

29 41.6 


10 

7766.8 

3921.9 


10 

9380.5 

5262,3 

20 

6514.3 

2945.9 


20 

7790.5 

3940.9 


20 

9411.3 

5288.6 

30 

6533 4 

2960,3 


30 

7814.3 

3960.1 


30 

9442.2 

5315,<» 

40 

6552 6 

2974.7 


40 

7838-I 

3979.4 


40 

9473 2 

5341.5 

50 

6571.9 

2989.2 


50 

7862.1 

3998.7 


50 

9504.4 

5368.2 

98 

6591.2 

3003.8 


108 

7886.2 

4018.2 


118 

9535.7 

5395.1 

10 

6610 6 

3018.4 


10 

7910.4 

4037.8 


10 

9567.2 

5422.1 

20 

6630.1 

3033.1 


20 

7934.6 

4057.4 


20 

9598 9 

5449.2 

30 

6649 6 

3047.9 


30 

7950.0 

4077,2 


30 

9630.7 

5476.5 

40 

6669.2 

3062.8 


40 

7983 5 

4097.1 


40 

9662.6 

5504.0 

50 

6688.8 

3077.7 


50 

8008.0 

4117.0 


50 

9694.7 

5531,7 

99 

6708.6 

3092.7 


109 

8032.7 

4137.1 


119 

97 27.0 

5559,4 

10 

67'28.4 

3107.7 


10 

8 )57 4 

4157.3 


10 

9759 4 

5587.4 

20 

6748.2 

3122 9 


20 

8082.3 

4177.5 


20 

9792.0 

5615.0 

30 

6768.1 

3138.1 


30 

8107.3 

4197.9 


30 

9824 8 

5613.8 

40 

67.S3.1 

3153.3 


40 

8132 3 

4218.4 


40 

9857.7 

5672.3 

50 

6808 2 

3168.7 


50 

8157.5 

4239.0 


50 

9890.8 

5700.9 

100 

6828.3 

3184.1 


no 

8182.8 

4259.7 


1*20 

9924.0 

5729.7 

10 

6848.5 

3199.6 


10 

8208 2 

4280.5 


10 

9957.5 

5758.6 

20 

6868.8 

3215.1 


20 

8233.7 

4301.4 


20 

9901.0 

5787.7 

30 

6889 2 

3230,8 


30 

8259.3 

4322 4 


30 

10025 0 

5817.0 

40 

6909.6 

3246.5 


40 

8285.0 

4343.6 


40 

10059 0 

5846,5 

50 

6930,1 

3262.3 


50 

8310.8 

4364.8 


50 

10093.0 

5876.1 


n 




































106 


A MANUAL OF LAND SURVEYING. 


CURVE FORMULAE. 


Chord def.=-chords2 

R 

No. chords= T 

« 

D 

Tan. def.=*^ chord.def. 

The square of any distance, divided by twice the ra¬ 
dius, will equal the distance from tangent to curve, 
very nearly. 

Table XX contains Tangents and Externals to a 1° 
curve. Tan. and Ext. to any other radius may be found, 
nearly enough, by dividing the Tan. or Ext. opposite the 
given Central Angle by the given degree of curve. 

To find Beg. of Curve, having the central Angle and 
Tangent: Divide Tan. opposite the given Central Angle 
by the given Tangent. 

To find Deg. of Curve, having the Central Angle and 
External: Divide Ext. opposite the given Central Angle 
by the given External. 

To find Nat. Tan. and Nat* Ex. Sec. for any angle by 
Table XX : Tan, or Ext. of twice the given angle divid¬ 
ed by the radius of a 1° curve will be the Nat. Tan. or 
Ex. Sec. 

To find angle for a given distance and deflection. 

Rule 1. Multiply given distance by .01745 (del for 1° 
for 1 ft.), and divide given deflection by the product. 

Rule 2. Multiply given deflection by 57.3, and divide 
the product by the given distance. 

To find deflection for a given angle and distance: 
Multiply the angle by .01745, and the product by the dis¬ 
tance. 


T=R tan. 1 

T= 50 tan. % 1 
Sin. D 

Sin. D=50 
R 

Sin. D=50 tan. % I 
T 


R=T cot. % I 
R— 50 

Sin. P 

E=R ex. sec. 1 
E=T tan. x 4 I 










APPENDIX. 


DETERMINATION OF THE AZIMUTH OF POLARIS AND 

TRUE MERIDIAN AT ANY HOUR, THE STAR BEING 

VISIBLE, AND THE CORRECT LOCAL MEAN TIME BEING 

KNOWN. 

. [From U. S. Surveying Instructions, 1894.] 

In this article it is proposed to present a method,with 
two new and compact tables adapted to common clock 
time, with such plain directions for use that any person 
of ordinary intelligence can understand and apply them. 

As the surveyor should have a perfectly clear idea of 
what is meant by Astronomical Time (used to simplify 
computations), and the Hour Angle of Polaris, these 
terms will now be explained. 

The Civil Bay, according to the customs of society, 
commences at midnight and comprises twenty-four 
hours from one midnight to the next following. The 
hours are counted from 0 to 12 from midnight to noon, 
after which they are again reckoned from 0 to 12 from 
noon to midnight. Thus the day is divided into two 
periods of 12 hours each; the first of which is marked 
a. m., the last p. m. 

The Astronomical Day commences at noon on the civil 
day of the same date. It also comprises twenty-four 
hours; but they are reckoned from 0 to 24, and from the 
noon of one day to that of the next following. 

The civil day begins twelve hours before the astro¬ 
nomical day; therefore the first period of the civil day 
answers to the last part of the preceding astronomical 
day, and the last part of the civil day corresponds to the 
first part of the astronomical day. Thus, January 9, 2 
o’clock p. m., civil time, is also January 9, 2 h , astro¬ 
nomical time; and January 9, 2 o’clock a. m., civil time, 
is January 8, 14 h , astronomical time. 

The rule, then, for the transformation of civil time 
into astronomical time is this: If the civil time is marked 

L 107 J 



108 


A MANUAL OF LAND SURVEYING. 


* 

p. m., take away the designation p. m., and the astronomical 
time is had without f urther change; if the civil time is marked 
a . m., take one from the day and add twelve to the hours , re¬ 
move the initials a. m ., and the result is the astronomical time 
wanted. 

The substance of the above rule may be otherwise 
stated, as follows: When the surveyor takes an observa¬ 
tion during p. m. hours, civil time, he can say: the as¬ 
tronomical time is the hours and minutes passed since 
the noon of this day, and when observing in the a. m. 
hours, he can say the astronomical time is the hours and 
minutes elapsed since the noon of yesterday , in either 
case omitting the designation a. m. or p. m., and writ¬ 
ing for the day of the month, that civil date on which 
the noon falls, from which the time is reckoned. Fi¬ 
nally, the astronomical time may be called the hours and min¬ 
utes elapsed since the noon last passed, the astronomical 
date being that of the civil day to which the noon belongs- 
Thus, April 23, 4:15 p. m., civil time, is April 23, 4 h 15 m , 
astronomical time, and April 23, 4:15 a.m., civil time, is 
April 22, 16 h 15 m , astronomical time. 

The surveyor should thoroughly master this trans¬ 
formation of the civil time into astronomical time, as 
it will be the first duty he will have to perform after 
observing Polaris out of the meridian. 

The change can always be made mentally, no written work being re¬ 
quired. Table I might be easily altered to give the times by the civil 
count marked a. m. and p. m., but such an arrangement would greatly 
extend and complicate the following rules and examples, and correspond¬ 
ingly increase the chances for making mistakes. 

Hour Angle of Polaris. — In Fig. 2, Plate I, the full ver¬ 
tical line represents a portion of the meridian passing 
through the zenith Z (the point directly overhead), and 
intersecting the northern horizon at the north point N, 
from which, for surveying purposes, the azimuths of 
Polaris are reckoned east or west. The meridian is 
pointed out by the plumb line when it is in the same 
plane with the eye of the observer and Polaris on the 
meridian, and a visual representation is also seen in the 
vertical wire of the transit, when it bisects the star on 
the meridian. 












110 A MANUAL OF LAND SURVEYING. 

When Polaris crosses the meridian it is said to cul¬ 
minate; above the pole (at S), the passage is called the 
Upper Culmination , in contradistinction to the Lower 
Culmination (at S'). 

In the diagram,— which the surveyor may better un¬ 
derstand by holding it up perpendicular to the line of 
sight when he looks toward the pole,— Polaris is sup¬ 
posed to be on the meridian, where it will be about noon 
on April 10th of each year. The star appears to revolve 
around the pole in the direction of the arrows, once in 
every 23 h 56 m 4 s . 09 of mean solar time; it consequently comes 
to and crosses the meridian, or culminates , nearly four 
minutes earlier each successive day. The apparent mo¬ 
tion of the star being uniform, one quarter of the circle 
will (omitting fractions) be described in 5 h 59 m , one half 
in ll h 58 m , and three quarters in 17 h 57 m . For the posh 
tions s n s 2 , s 3 , etc., the angles SPs 15 SPs 2 , SPs 3 , etc., 
are called Hour Angles of Polaris for the instant the star 
is at Sj, s 2 , or s 3 , etc., and they are measured by the arcs 
Ss x , Ss 2 , Ss 3 , etc., expressed (in these instructions) in mean 
solar (common clock) time, and are always counted from 
the upper meridian (at S), to the west , around the circle 
from 0 h 0 m to 23 h 56 m .l, and may have any value between 
the limits named. The hour angles, measured by the 
arcs Ss 1} Ss 2 , Ss 3 , Ss 4 , Ss 5 , and Ss 6 , are approximately 
l h 8 m , 5 h 55 m , 9 h 4 m , 14 h 52 m , 18 h 01 m , and 22 h 48 m respect¬ 
ively; their extent is also indicated, graphically, by 
broken fractional circles about the pole. The hour an¬ 
gle, 5 h 55 m and 18 h 01 m are those at west and east elonga¬ 
tion, respectively, in latitude 40° N. 

Suppose the star observed (e. y.) at the point S 3 ; the 
time it was at S (the time of upper culmination), taken 
from the whole circle, 23 h 56 ra .l, will leave the arc Ss,, 
s 2 , s 3 , or the hour angle at the instant of observation; 
similar relations will obtain when the star is observed 
in any other position; therefore, in general: — 

Subtract the time of Upper Culmination from the 
correct local mean time of observation; the remainder 
will be the Hour Angle of Polaris. 

The observation will be made as heretofore directed, 



TO FIND A TRUE MERIDIAN. 


Ill 


modified as follows: There will be no waiting for the 
star to reach elongation; the observation maybe made 
at any instant when Polaris is visible, the exact time 
being carefully noted. 


Table I. 


This table gives, in “ Part I,” the local mean time of 
the upper culmination of Polaris, on the 1st and 15th of 
each month, for the years 1890 to 1900, inclusive. The 
times decrease, in each year, to April 10, when they be¬ 
come zero; then, commencing at 23 h 56 m .l, the times 
again decrease until the following April, and so on, con¬ 
tinuously. The quantity in the column marked “ Piff. 
for 1 day” is the decrease per day during the interval 
of time against which it stands, and answers for all the 
years marked in the table. For any intermediate date, 
the “Diff. for 1 day” will be multiplied by the days 
elapsed since the preceding tabular date, and the prod¬ 
uct subtracted from the corresponding time, to obtain 
the required time of upper culmination for the date 
under consideration. The table answers directly for 90° 
west longitude. For places east or west of the assumed 
meridian, a small correction, dependent on the longi¬ 
tude, may be applied to the deduced time of culmina¬ 
tion. The correction for longitude should not be used 
for dates subsequent to Dec. 31, 1896. This correction 
may be taken from Part III, and, with sufficient accu¬ 
racy, for the longitude nearest that of the station. Use 
the correction according to the direction placed over it. 
A few examples will illustrate the use of the table. 


1. Required, the time of upper culmination of Polaris for a station in 
longitude 116° west, for March 3, 1892. 

h. m. 

Astron. time, U. C. of Polaris, 1892, March 1. 2 37.8 

Red. for 2 days is 3“.9JX2=7 m .9 (Part II) l Subtract 8 2 

Cor. for 116° long, is.0“.3 (Part III) f bUDtrdCt . 

Local mean time, U. C. of Polaris, 1892, March 3. 2 29.6 


The required time may also be obtained by using the 
table in the opposite direction; by taking the time for 
March 15. and adding the reduction, as follows: — 







112 


A' MANUAL OF LAND SURVEYING. 


h. in. 

Astron. time, U. C. of Polaris, 1892, March 15. 1 42.6 

Red. for 12 days is 3 m .94Xl2=47 m .3, add. 47.3 


Sum. . 2 29.9 

Correction for longitude 116° (Part III), subtract. 0.3 

Local mean time, U. C. of Polaris, 1892, March 3. 2 29.6 

In this case the two results are identical. If the 
computation is made both ways, the results will check 
each other. 

Part II has been inserted to save the surveyor the lit¬ 
tle trouble of making multiplications; thus, for the 
above example, look in Part II, under the proper tabu¬ 
lar difference, 3 m .94, and opposite the day of the month 
in left hand column is the correction 7 m .9; also in Part 
III is found the correction for 116° longitude, 0 ra .3, the 
sum being 8 m .2. The work may be put down as follow’s:— 


h. m. 

Astron. time, U. C. of Polaris, 1892, March 1 (Part I). 2 37.8 

Red. (Part II), and correction for long. (Part III), subtract. 8.2 


Local mean time, U. C. of Polaris, 1892, March 3. 2 29.6 


The longitude correction being small, may generally be 
omitted; it will not be considered in the following ex¬ 
amples. 

Computing from a preceding date, for days between 
April 11 and 15 of any year, the reduction in Part II 
will be greater than the tabulated time of culmination, 
in which case 23 h 56 m .l will be added, to make the sub¬ 
traction possible. 

2 . Required, for a station in long. 90° west, the time of U. C. of Polaris 


for April 14, 1891: — 

h. m. 

Astron. time, U. C. of Polaris, 1891, April 1 (Part I). 0 38.4 

Add...... 23 56.1 

Sum . . 24 34.5 

Reduction to April 14 (Part II), subtract. 51.1 

Local mean time, U. C. of Polaris, April 14. 23 43.4 


Working from a follcncing date, for days between the 
9th and 15th of April, the sum will exceed 23 h 56 m .l, 
and when this occurs subtract 23 h 56 m .l from the sum, 
and the remainder will be the required time. 



















TO FIND A TRUE MERIDIAN. 113 


3. Required, for a station in long. 90° west, the time of U. C. of Polaris 
for April 10, 1892: — 

Astron. time, U. C. of Polaris, 1892, April 15 (Part I). 23 36.8 

Reduction for 5 days (Part II), add . 19.6 


Sum 

Subtract 


23 56.4 
23 56.1 


Local mean time, U. C. of Polaris, 1892, April 10. 0 0.3 


This example, worked like the last one, from the pre¬ 
ceding date (April 1), will give precisely the result above 
written. (See example above.) If to the above time of 
culmination we add 23 h 56 m .l, and then subtract 3 m .9, we 
obtain 23 h 52 m .5, the time of the second upper culmina¬ 
tion on April 10, since both occur within 24 hours of 
noon and consequently on the same day. The upper cul¬ 
mination, to be used at any time, will always be the 
last one that occurs before the observation. In this in¬ 
stance it is, of course, the first one that takes place on 
the 10th. The second culmination occurs 7 m .5 before noon 
of April 11, and consequently in broad daylight. 

The surveyor should be careful to employ Part II, 
Table I, correctly. When the table is used in regular 
order, the “Reduction” may be taken from Part II 
with the argument,! “Pay of the month ” in left hand 
column, or, “Number of days elapsed” in right hand 
column, as may be preferred. In example 2, Part II, 
may be entered in with the argument 13 days elapsed 
(from 1st to 14th) in right hand column; then the reduc¬ 
tion, 51 m .l, results, as above written; but, when work¬ 
ing from a following date (example 3), the day of the 
month in left hand column cannot be used. 

Mistakes are often made by using the wrong column 
in Part I; as a matter of course, the time should always 
be taken out for the current year. 

The foregoing examples embrace all cases which can 
occur in the use of Table I, and will be a sufficient guide 
for its application. 


+ “ Argument,” the quantity on which another quantity in a tabic 
depends. 









114 


A MANUAL OF LAND SURVEYING, 


Table I. 

Local mean ( astronomical) time of the upper culmination of 
Polaris , computed for longitude 6 hours (90°) 
west of Greenwich. 

[The time on line with any date in Parti is the hours and minutes elapsed 
(measured, by a common clock or watch) since the preceding noon.] 


Pf(rt I. 


Date. 

1895. 

1896. 

1897. 

1898. 

1899. 

1900. 

Ditf. 

for 

1 day 


h. to. 

h. m. 

h. to. 

h. to. 

h. TO. 

h. to. 

m. 

Jan. 1 

6 34.7 

6 36.1 

6 33.0 

6 34.1 

6 35.2 

6 36.3 

3.95 

15 

5 39.4 

5 40.8 

5 37.7 

5 38.8 

5 39.9 

5 41.0 

3.95 

Feb. 1 

4 32.3 

4 33.7 

4 30.6 

4 31.7 

4 32.8 

4 33.9 

3.95 

15 

3 37 .1 

3 38.5 

3 35.3 

3 36.4 

3 37 5 

3 38.6 

3 95 

Mar. 1 

2 41.8 

2 39.3 

2 40.1 

2 41.2 

2 42.3 

2 43.4 

3.94 

15 

1 46.7 

1 44.1 

1 44.9 

1 46.0 

1 47.1 

1 48.2 

3.94 

April 1 

0 39 7 

0 37.2 

0 38.0 

0 39.1 

0 40.2 

0 41.3 

3.94 

15 

23 40 8 

23 38.3 

23 39.1 

23 40.2 

23 41.3 

23 42.4 

3.93 

May 1 

22 38.0 

21 35.5 

22 36.2 

22 37.3 

22 38.4 

22 39.5 

3.93 

15 

21 43.0 

21 40.6 

21 41.3 

21 42.4 

21 43 5 

21 44.6 

3.91 

June 1 

20 36.4 

20 33.9 

20 34.7 

20 35.8 

20 36.9 

20 38 0 

3 92 

15 

19 41.6 

19 39.1 

19 39.9 

19 41.0 

19 42.1 

19 43.2 

3 92 

July 1 

18 38.9 

18 36.5 

18 37.2 

18 38.3 

18 39.4 

18 40 5 

3.92 

15 

17 44.1 

17 41.7 

17 42.4 

17 43.5 

17 44.6 

17 45.7 

3.92 

Aug. 1 

16 37.6 

16 35.1 

16 35.8 

16 36.9 

16 38.0 

16 39 1 

3.91 

15 

15 42.7 

15 40.3 

15 41.0 

15 42.1 

15 43.1 

15 44.3 

3.92 

Sept. 1 

14 38.1 

14 33.7 

14 31.3 

14 35.4 

14 46.5 

14 37.6 

3.92 

15 

13 41.2 

13 38.8 

13 39.4 

13 40.5 

13 41.6 

13 42.7 

3.92 

Oct. 1 

12 38.4 

12 36.0 

12 36.6 

12 37.7 

12 38.8 

12 39 9 

3.93 

15 

11 43 4 

11 41.0 

11 41.6 

11 42.7 

11 43.8 

11 44.9 

3.93 

Nov. 1 

10 36 6 

10 34.1 

10 34.8 

10 35.9 

10 37.0 

10 38.1 

3.93 

15 

9 41.5' 

9 39.0 

9 39 6 

9 40.7 

9 41.8 

9 42 9 

3.94 

Dec. 1 

8 38.4 

8 35.9 

8 36.6 

8 37.7 

8 38.8 

8 39 9 

3.94 

15 

7 43.2 

7 40.7 

7 41.4 

7 42.5 

7 43.6 

7 44.7 

3.94 

3.95 


Part II. —Reduction of tabular times to intermediate dates. Subtract the 
reduction when computing from a preceding , or add it when work¬ 
ing from a following date. 



Reduction. Arg. — "Ditf. for 1 day.” 

No. of 

Day of the month. 

m. 

3.91. 

to. 

3.92. 

TO. 

3.93. 

TO. 

3.94, 

TO. 

3.95. 

days 

elaps’d. 

2 or 16. 

TO. 

3.9 

TO. 

3.9 

TO. 

3.9 

TO. 

3.9 

TO. 

3.9 

1 

3 or 17 . 

7.8 

7.8 

7.9 

7.9 

7.9 

O 

IS* 

4 or 18 . 

11.7 

11.8 

11.8 

11.8 

11.8 

3 

5 or 19. 

15.6 

15.7 

15.7 

15.8 

15.8 

4 

6 or 20 . 

19.5 

19.6 

19.6 

19.7 

19.7 

5 

7 or 21 . 

23 5 

23.5 

23.6 

23.6 

23.7 

6 

8 or 22 . 

27.4 

27.4 

27.5 

27.6 

27.6 

7 

9 or 23 . 

31.3 

31.4 

31.4 

31.5 

81.6 

8 

10 or 24 . 

35.2 

35.3 

35.4 

35.5 

35.5 

9 

11 or 25 . 

39.1 

39.2 

39.3 

39.4 

39.5 

10 

12 or 26 . 

43.0 

43.1 

43.2 

43.3 

43.4 

11 

13 or 27. 

47.0 

47.0 

47.2 

47.3 

47.4 

12 

14 or 28. 

50.8 

51.0 

51.1 

51.2 

51.3 

13 

29. 

54.7 

54,9 

55.0 

55.2 

55.3 

14 

30. 

58.6 

58.8 

58.9 

59.1 

59.2 

15 

31. 

62.6 

62.7 

62.9 

63.0 

63.2 

16 





























































TO FIND A TRUE MERIDIAN. 


115 


Applications of Tables I and II. 

4. Required the Hour Angle and Azimuth of Polaris , for a station in 
latitude 46° N., longitude 90° W., at 8 h 24 m p. m., November 7, 1891. 


h. m. 

Astronomical time of observation, 1891, Nov. 7. 8 24.0 

h. m. 

Astron. time, U. C. Polaris, Nov. 1 (TableI, Parti), 10 35.1 
Reduction to Nov. 6* * * § (Part II), subtract. +19.7 

Astron. time, U. C. Polaris, Nov. 6.10 15.4, subt. £10 15.4 

Hour Angle of Polaris, at observation. 22 8.6 

Subtract from.§23 56.1 

Time Argument for Table II. 1 47 5 

Azimuth of Polaris, at observation.1° 51' E. 


Part III. — Correction of the tabular time for longitude. 


Longitude. 

63° 

72° 

81° 

90° 

99° 

108° 

117° 

127° 


Add 

Add 

Add 

Add 

Subt. 

Subt. 

Subt. 

Subt. 


m. 

m. 

m. 

m. 

m. 

m. 

m. 

m. 

Correction. 

0.3 

0.2 

0.1 

0.0 

0.1 

0.2 

0.3 

0.4 . 


5. Required the Hour Angle and Azimuth of Polaris , for a station in 
latitude 41° 12' N., longitude 94° W., at 6 k 16 m a. m., Nov. 19, 1898. 


h. m. 

Astronomical time of observation, 1898, Nov. 18.18 16.0 

h. m. 

Astron. time, U. C. Polaris, Nov. 15 (Table I, Part I), 9 40.7 
Reduction to Nov. 19 (Part II), subtract. 15.8 

Astron. time, U. C. Polaris, Nov. 19.9 24.9, subt.9 24.9 

Hour Angle of Polaris, at observation, and Time Argument for 

Table II.»8 51.1 

Azimuth of Polaris , at observation (Table II).11° 11' W. 


*By reference to the above table, the surveyor will observe that the 
times, between Nov. 1 and 15, are greater than 8 h 24 m ; consequently, the 
culmination for one day earlier, Nov. 6, will be used; see directions on 
page 111; also, last clause of example 3, page 112. 

+ From Part II. Table I, opposite 6th day of month, and under “3 94 m .” 

+ To subtract, take 1 day from Nov. 7, and add its equivalent, 24 h , to 8 h 
24 m , making, Nov. 6, 32 h 24 m (which is the time expressed by Nov. 7, 8 h 
24 m ); then subtract in the usual manner. 

§ See last clause of footnote, page 115. 

|| In case the Hour Angle comes out greater than ll h 58™, subtract it from 
23 h 56.l m ; see example 4, on above. 

1 The Hour Angle being less than ll h 58 m , the Azimuth is west; see pre¬ 
cepts, top of Table II. 









































116 


A MANUAL OF LAND SURVEYING. 


Table II. 

This table gives, for various hour angles, expressed in 
mean solar time , and for even degrees of latitude from 30 
to 50 degrees, the Azimuths of Polaiis during the remain¬ 
der of this century, computed for average values of the 
north polar distance of the star — the arguments (refer¬ 
ence numbers), being the hour angle (or 23 h 56 m .l, minus 
the hour angle, when the latter exceeds ll h 58 m ), which 
is termed the Time Argument; and the latitude of the 
place of observation. The table is so extended that 
azimuths may be taken out by mere inspection, and all 
interpolation avoided, except such as can be performed 
mentally. 


The vertical diameter SS\ Plate I, Fig, 2, divides the apparent path of 
Polaris into two equal parts, and for the star at any point s 6 on the east 
side, there is a corresponding point s,, on the ivest side of the meridian, 
for which the azimuth Nw is equal to the azimuth Ne. The arc Ss l S's R , 
taken from the entire circle (or 23 h 56 m .l), leaves the arc Ss 6 . and its equal, 
SSj, expressed in time, may be used to find, from Table II, the azimuth 
Nw, which is equal to Ne. 

The hour angles entered in Table II include only those of the west half 
of the circle ending at S', and when an hour angle greater than ll h 58 m re¬ 
sults from observation, it will be subtracted from 23 h 56 m .l, and the re¬ 
mainder will be used as the “ time argument ” for the table. The surveyor 
should not confound these two quantities. The hour angle itself always 
decides the direction of the azimuth and defines the place of the star with 
reference to the pole and meridian, as noted at top of Table II. See ex¬ 
amples below Table I, page 114. 

The hours of the “ time arguments ” are placed in the 
columns headed “Hours,” on left of each page. The 
minutes of the time arguments will be found in the col¬ 
umns marked “m.,” under the years for which they are 
computed, and they are included between the same 
heavy zigzag lines which inclose the hours to which 
they belong. 

The time arguments are given to the nearest half 
minute; the occurrence of a period after the minutes of 
any one of them, indicates that its value is 0.5 m greater 
than printed, the table being so arranged to economize 
space. 


TO FIND A TRUE MERIDIAN. 


117 


, The tables will be used as follows: Find the hours of 
the time argument in the left-hand column of either 
page; then, between the heavy lines which inclose the 
hours, find the minutes in the column marked at the top 
with the current year. On the same horizontal line 
with the minutes, the azimuth will be found under the 
given latitude, which is marked at the top of the right- 
hand half of each page. Thus, for 1892, time argument, 
0 h 40 m , latitude 42°; find 0 h on left-hand page and under 
1892, find 40 m , on tenth line from the top, and on same 
line with the minutes , under latitude 42°, is the azimuth 
0° 18'. For 1896, time argument 7 h 58 m , lat. 36°, the azi¬ 
muth is 1° 19', found on the 9tli line from bottom of 
right-hand page. 

If the exact time argument is not found in the table, 
the azimuth should be proportioned to the difference 
between the given and tabular values of said argument. 
Thus, if the time argument in the first of-the above ex¬ 
amples (for 1892) was 0 h 42 m , instead of 0 h 40 m , the azi¬ 
muth would be the mean between 0° 18' and 0° 20', or 0° 
19'. In a similar manner, if the latitude is nearer an odd 
than an even degree, the mean of the azimuths for the 
next greater and next less latitude will be used; thus, 
in the above example for 1896, if the given latitude was 
37°, the mean between 1° 19' and 1° 21', or 1° 20', would 
be the corresponding azimuth. The table has been ar¬ 
ranged to give the azimuths as exemplified above, by 
simple inspection. No written arithmetical work is re¬ 
quired, all being performed mentally. It will always be 
sufficient to take the nearest whole degree of latitude 
and use it as above directed, except for a few values 
near the bottom of either page, where the difference of 
azimuths, for 2° difference of latitude, amounts to4or 5 
minutes of arc; for example, 1890, time argument, 7 h 
29 m , lat. 46° 41'. In this case the latitude may be taken 
to the nearest half degree (461°); the corresponding azi¬ 
muth is 1° 42'. 

3. The attention of the surveyor is directed to the 
fact that he should always use one day of twenty-four hours 
as the unit when he subtracts the time of culmination 
33 







118 


A MANUAL OF LAND SURVEYING. 


from the time of observation. See example 4, page 114. 
In any case when the time of upper culmination, taken 
from Table I, for the given date, would be numerically 
greater than the astronomical time of observation, the 
former time will be taken out for a date one day earlier 
than the date of observation. The surveyor will decide 
when such condition exists by comparing the time given 
in the table with his astronomical time of observation. 
See example 4 and explanations in footnotes below 
Table I, page 113. 

When an hour angle comes out within one minute of 
either 0 h 0 ra , or 23 h 56 m .l, the observation may be regarded 
as having been taken with the star on the meridian. 
above the pole; if within one minute of ll h 58 m , Polaris 
may be considered on the meridian belcnv the pole at the 
time of observation. 

At elongation Polaris is nearly 5 h 55 m west (or east) of 
its position at upper culmination; consequently if the 
hour angle for any observation comes out within Jive 
minutes of 5 U 55 m or 18 h l m , the star may be assumed to 
be at elongation, west for the first and east for the second 
hour angle, and its azimuth may be taken from a pre¬ 
ceding table, which gives its value at elongation, from 
1890 to 1905, inclusive. 

Should the surveyor wish the time of Lower Culmina¬ 
tion, for use with the plumb-line method, described on 
page 32, or for any other purpose, he will first deter¬ 
mine the time of upper culmination for the date (Table 
I), and then subtract ll h 58 m for the preceding lower cul¬ 
mination, or add ll h 58 m for the lower culmination 
following the derived time for upper culmination, at¬ 
tending to the addition or subtraction of 23 h 56 m .l, as 
directed on page 111. 

The time to be used when making observations on 
Polaris off the meridian, should be as accurate as can be 
obtained. Looking at Table II, near the top of either 
page, the surveyor will observe, that for a difference of 
four minutes in the time argument, there is a change of 
about two minutes in azimuth; consequently, to obtain 
the azimuth to the nearest whole minute of arc, the local 





TO FIND A TRUE MERIDIAN. 


119 


mean time , upon which all depends, should be known 
within two minutes. When the surveyor uses a solar in¬ 
strument, he can readily determine the time for him¬ 
self during the afternoon before observing Polaris, or in 
the morning after observation, and, without moving the 
hands of his watch, apply the necessary correction to 
his observed watch time. When the surveyor uses stan¬ 
dard railroad time , he will correct the same for the differ¬ 
ence of longitude between his station and the standard 
meridian for which the time is given, at the rate of 
four minutes of- time for each degree of the difference in 
arc. Thus, if the difference of longitude is 6° 45', the 
equivalent in time will be 27 minutes. The difference 
of longitude may be taken from a good map. The num¬ 
ber of seconds taken from Table III, multiplied by the 
number of ranges, will give the correction for longitude 
in seconds of time. The correction will be subtracted from 
the standard railroad time of observation, when the 
surveyor’s station is west , or added when east of the stan¬ 
dard meridian, as the case may require, to obtain local 
time. It is immaterial where the surveyor obtains the 
standard time, provided he gets it right; a result which 
will be determined in the most satisfactory manner, 
by a direct comparison at telegraph office, personally 
conducted. 

Generally, the surveyor will have only two or three 
simple additions or subtractions to make, and ten min¬ 
utes will be ample time in which to make the observa¬ 
tion and perform the little computation required. 

Note.—T he azimuths entered in the following table 
were calculated with the mean North Polar Distance of 
Polaris (1° 16' 32'), the assumed latitudes of the table, 
and the stated hour angles for the year 1890. The result¬ 
ing values having been tabulated, the process was re¬ 
versed, and with the mean N. P. D. of the star, for the 
1st of July of each of the remaining ten years of the 
series, the latitudes named, and azimuths already deter¬ 
mined. the corresponding hour angles were found. 






Table IT.— Azimuths of Polaris for the use of land surveyors. 


[The hour angles are expressed in mean solar time. The occurrence of a 
period after minutes of an hour angle indicates that its value is 0 m .5 greater than 
printed.] 


Star an® Azimuth. 


W. of N. when hour angle is 
less than 1 l h 58 m . 

E. of N. when hour angle is 
greater than 1 1 >> 58 m . 


Time argument, the star’s 
hour angle (or 23 h 56 m .l 
minus the star’s hour an¬ 
gle), for the year — 


tfl 



• 

• 


0 



1C 

I'- 

X 

Ct 



a 

G 

C* 

Ci 

Ci 



x 

cc 

oc 

X 

<X) 

Cl 








h. 

in. 

m. 

m 

m. 

m. 

m. 

0 

4 

4 

4 

4 

4 

4 


8 

8 

8 

8 . 

8 . 

8 . 


12 

12 . 

12 

12 . 

12 . 

12 . 


16. 

16. 

16. 

16. 

16. 

16. 


20 . 

20 . 

20 . 

20 . 

21 

21 


24 

24. 

24. 

24. 

25 

25 


28. 

28. 

28. 

29 

29 

29 


32. 

32. 

33 

33 

33. 

33. 


36. 

37 

37 

37 

37. 

37. 


41 

41 

41 

41. 

41. 

41. 


45 

45 

45. 

45 

45. 

46 


49 

49 

49. 

49. 

50 

50 


53 

53. 

53 

53. 

54 

54 

0 

57 

57. 

57. 

58 

58 

58 

1 

1 . 

1 . 

2 

2 

o 

(V . 

9 


6 . 

6 

r* 

i 

ry 
i 

\r* 

4 . 

7. 


11 . 

11 . 

12 

12 

13 

13 


16. 

17 

17. 

17. 

18 

18 


21 . 

22 

22 . 

22 . 

23 

23. 


27 

27 

28 

28. 

28. 

28. 


32 

32. 

33 

33. 

34 

34 


37 

37. 

38 

38 

38. 

39 


42 

42. 

43 

43. 

44 

44. 


47 

47. 

48 

48. 

49 

49. 


52. 

53 

53. 

54 

54. 

55 

1 

67 

58 

58. 

59 

59. 

nr 

O 

2 . 

3 

3. 

4 

5 

5. 


8 

8 . 

9 

9. 

10 

10 . 


12 . 

13. 

14 

14. 

15 

16 


18 

18. 

19 

20 

20 . 

21 . 


23 

24 

24. 

25 

26 

26. 


28 

29 

30 

30 

31 

32 


33. 

34. 

35 

35. 

36. 

37 


38. 

39 

40 

41 

41. 

42. 


43. 

44. 

45. 

46. 

47 

48 


49 

49. 

50. 

51 . 

53 

53 


54 

55 

56 

57 

5 

58. 

2 

59 

0 

1 

2 

3 

4 

3 

4. 

5-1 

6 . 

7. 

8 . 

9. 


Polaris above the Pole. 


I 


To determine the true meridian, the azimuth 
will be laid off to the east when the hour 
angle is less than ll h 58 m , and to the west 
when greater than ll h 58 m . 


Azimuths for latitude 


o 

30 

o 

32 

o 

34 

o 

36 

o 

38 

© • 

4?) 

° 

42 

Q 

44 

o 

46 

o 

-IS 

o 

50 

o / 

o / 

o 

o / 

o / 

o / 

O / 

o / 

O / 

o / 

o / 

0 2 

0 2 

0 2 

C 2 

0 2 

0 2 

0 2 

0 2 

0 2 

0 2 

0 2 

3 

3 

3 

3 

3 

4 

4 

4 

4 

4 

4 

5 

5 

5 

5 

5 

5 

6 

6 

6 

6 

6 

6 

6 

6 

7 

7 

7 

7 

8 

8 

8 

8 

8 

8 

8 

8 

9 

9 

9 

9 

10 

10 

11 

9 

10 

10 

10 

10 

11 

11 

11 

12 

12 

13 

11 

11 

11 

12 

12 

12 

13 

13 

14 

14 

15 

12 

13 

13 

13 

14 

14 

15 

15 

16 

16 

17 

14 

14 

15 

15 

15 

16 

16 

17 

18 

. 18 

19 

15 

16 

16 

17 

17 

18 

18 

19 

20 

20 

21 

17 

17 

18 

18 

19 

19 

20 

21 

21 

23 

23 

19 

19 

19 

21* 

21 

22 

22 

23 

23 

24 

25 

2n 

21 

21 

22 

22 

23 

24 

24 

25 

26 

27 

99 

22 

23 

23 

24 

25 

25 

26 

27 

. 28 

29 

23 

24 

24 

25 

26 

26 

27 

28 

29 

30 

3' 

25 

26 

26 

21 

28 

28 

29 

30 

31 

33 

34 

27 

27 

28 

29 

30 

31 

32 

33 

34 

3 5 

37 

29 

29 

30 

31 

32 

33 

34 

35 

36 

3s 

3 ) 

31 

31 

32 

83 

34 

35 

36 

37 

38 

40 

42 

32 

33 

34 

35 

36 

37 

38 

39 

41 

42 

41 

34 

35 

36 

37 

38 

39 

40 

42 

43 

45 

47 

36 

37 

38 

39 

40 

41 

42 

44 

45 

47 

49 

38 

39 

40 

41 

42 

43 

44 

46 

47 

49 

51 

39 

40 

41 

42 

44 

45 

46 

48 

50 

52 

54 

41 

42 

43 

44 

46 

47 

48 

50 

52 

54 

56 

43 

44 

45 

46 

47 

49 

50 

52 

54 

56 

0 59 

43 

46 

47 

48 

49 

51 

52 

54 

56 

) 58 

1 1 

46 

47 

49 

50 

51 

53 

54 

56 0 58 

1 1 

3 

48 

49 

50 

51 

53 

54 

56 

0 58 

1 0 

3 

K 

o 

50 

51 

52 

53 

55 

56 

0 58 

1 0 

2 

5 

fi 

51 

52 

54 

55 

57 

0 58 

1 0 

9 

-V 

4 

4 

10 

53 

51 

55 

57 

0 58 

l 0 

2 

4 

6 

9 

12 

54 

55 

57 

0 58 

1 0 

9 

*v 

4 

6 

8 

11 

14 

50 

57 

0 58 

1 0 

9 

3 

6 

8 

10 

13 

16 

57 

0 59 

1 0 

•> 

Sm/ 

3 

5 

-*■ 

l 

9 

12 

15 

18 

0 59 

1 0 

2 

3 

5 

7 

9 

11 

14 

17 

20 

1 0 

*> 

3 

5 

7 

8 

11 

18 

16 

19 

22 

2 

3 

5 

6 

8 

10 

12 

15 

17 

20 

24 

3 

4 

6 

s| 

10 

12 

14 

16 

19 

22 

26 


1130 } 













































































TO FIND A TRUE MERIDIAN 


121 


Table II.— Continued. 


Stab and Azimuth. I Polap.is the Pole. 


Hours. 

1895. 

• 

C5 

x 

1897. 

X 

cs 

X 

C 

1 GO 

1900. 



Azimuths for latitude- 




o 

:J0 

o 

32 

o 

34 

o 

36 

o 

38 

Q 

40 

o 

42 

O 

44 

o 

46 

o 

48 

o 

50 

h. 

m. 

rn. 

m. 

m. 

m. 

m. 

o / 

o / 

o / 

o t 

o / 

O f 

o / 

o t 

o / 

o / 

o t 

3 

10 . 

12 

13 

14 

15 

16 

5 

6 

8 

10 

12 

13 

16 

18 

21 

24 

28 


17 

18 

19. 

20 . 

21 . 

23 

6 

8 

9 

u 

13 

15 

18 

20 

23 

27 

30 


23 

24. 

25. 

27 

28 

29 

8 

9 

11 

13 

15 

17 

19 

22 

25 

28 

32 


29. 

31 

32. 

33. 

35 

36 

9 

11 

13 

14 

16 

19 

21 

24 

27 

30 

34 


36 

37. 

39 

40 

41. 

43 

11 

13 

14 

16 

18 

20 

23 

25 

29 

32 

36 


43. 

45 

46 

48 

49 

51 

12 

14 

16 

17 

20 

22 

25 

27 

31 

34 

38 


51 

52. 

1 54 

55 

57. 

59 

14 

15 

17 

19 

21 

24 

26 

29 

32 

36 

40 

3 

58. 

0 

2 

3. 

5 

7 

15 

17 

19 

21 

23 

25 

28 

31 

34 

38 

42 

4 

8 

10 

12 

13. 

15. 

17. 

17 

19 

21 

23 

25 

27 

30 

33 

36 

40 

44 


19 

21 . 

23. 

25. 

28 

30 

19 

21 

23 

24 

27 

29 

32 

35 

39 

43 

47 


30. 

33 

35. 

38 

40. 

43 

20 

22 

24 

26 

29 

31 

34 

37 

41 

45 

49 

4 

4 <5 

45 

48 

50. 

54 

57. 

22 

24 

26 

28 

30 

33 

36 

39 

42 

47 

51 

5 

0 . 

4. 

8 . 

12 . 

17. 

23 

24 

26 

28 

30 

38 

35 

38 

41 

45 

49 

54 


20 . 

29 

37 

50. 



26 

27 

30 

32 

34 

37 

40 

43 

47 

51 

53 








27 

29 

31 

33 

36 

39 

42 

45 

49 

53 

58 

5 

.... 

• • 





1 29 

1 30 

1 32 

1 35 

1 37 

1 40 

1 43 

1 47 

1 50 

1 55 

1 59 

Star and Azimuth. 



Polaris beloiv the 

Pole. 



11 

54 

54 

54 

54 

54 

54 

10 1 

0 10 1 

0 2 

0 2 

0 2 

0 2 

0 2 

0 2 

0 2 

0 2 


50 

50 

50 

50 

50 

50 

3 

3 

3 

3 

3 

3 

3 

4 

4 

4 

4 


46 

46. 

45. 

45. 

45. 

45. 

5 

5 

5 

5 

5 

5 

5 

5 

6 

6 

6 


42 

41 

41. 

41. 

41. 

41. 

6 

6 

6 

6 

7 

7 

7 

7 

8 

8 

8 


37 . 

37. 

37. 

37 . 

37. 

37 

8 

8 

8 

8 

8 

8 

9 

9 

9 

10 

10 


33 . 

33 . 

33 . 

33 . 

33 

33 

9 

9 

9 

10 

10 

10 

11 

U 

11 

12 

12 


29. 

29. 

29. 

29 

29 

29 

11 

11 

11 

11 

12 

12 

12 

13 

13 

14 

14 


25. 

25 

25 

25 

25 

25 

12 

12 

13 

13 

13 

14 

14 

14 

15 

15 

16 


21 

21 

21 

21 

21 

21 

14 

14 

14 

15 

15 

15 

16 

16 

17 

17 

18 


17. 

17 

17 

17 

16. 

16. 

15 

15 

16 

16 

17 

17 

18 

18 

19 

19 

20 


13. 

13 

13 

12 . 

12 . 

12 . 

17 

17 

17 

18 

18 

19 

19 

20 

21 

21 

22 


9. 

9 

8 . 

8 . 

8 . 

8 

18 

18 

19 

19 

20 

20 

21 

22 

22 

23 

25 


5. 

5 

4. 

4. 

4. 

4 

20 

20 

20 

21 

22 

22 

23 

23 

24 

25 

26 

11 

1 

1 

1 

0 . 

0 . 

0 

21 

21 

22 

23 

23 

24 

24 

25 

26 

27 

28 

10 

56. 

56. 

56. 

56. 

55. 

55. 

23 

23 

24 

24 

25 

25 

26 

27 

28 

29 

30 


51. 

51 . 

51 

51 

50. 

50. 

24 

25 

25 

26 

27 

27 

28 

29 

30 

31 

32 


46. 

46. 

46 

46 

45. 

45 

26 

27 

27 

28 

29 

29 

30 

31 

32 

34 

35 


41. 

41. 

41 

40. 

40. 

40 

28 

29 

29 

30 

31 

31 

32 

33 

35 

36 

37 


36. 

36 

35. 

35. 

35 

35 

30 

30 

31 

32 

33 

34 

35 

36 

37 

38 

40 


31. 

31 

30. 

30. 

30 

29. 

32 

32 

33 

34 

35 

36 

37 

38 

39 

41 

42 


26 

26 

25. 

25 

24. 

24. 

33 

34 

35 

36 

37 

38 

39 

40 

41 

43 

44 


21 

21 

20 . 

20 

19. 

19 

35 

36 

37 

37 

39 

40 

41 

42 

43 

45 

47 


16 

15. 

15 

15 

14. 

14 

37 

38 

39 

39 

40 

41 

43 

44 

46 

47 

49 

10 

11 

10 . 

10 . 

9. 

9 

8 . 

39 

39 

40 

41 

42 

43 

45> 

46 

48 

49 

51 


I 



























































































122 


A MANUAL OF LAND SURVEYING, 


Table II.— Concluded. 


Star and Azimuth. 



• 

Polaris below the 

Pole. 



• 









Azimuths for latitude- 




ft* 

S3 

to 

cs 

• 

«o 

es 

r- 

a 

OD 

os 

© 

© 

o 

o 

O 

o 

o 

Q 

o 

O 

o 

o 

o 

O 

= 

2 

cc 

<x> 

a r. 

QC 

rH 

c: 

30 

32 

34 

36 

38 

40 

42 

44 

46 

48 

50 

h. 

m. 

m. 

m. 

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1 21 

1 30 

1 32 

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1 37 

1 40 

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1 47 

1 50 

1 55 

1 59 


Table III.— Difference of Longitude of Meridians six miles apart 

in Seconds of time. 


Lat. 

Seconds. 

Lat. 

Seconds. 

Lat. 

Seconds. 

Lat. 

Seconds. 

O 


O 


O 


O 


30 

24.02 

35 

25.40 

40 

27.14 

45 

29.39 

31 

24.27 

36 

25 71 

41 

27.55 

46 

29.92 

32 

24.53 

37 

26.04 

42 

27.97 

47 

30.47 

33 

24.80 

38 

26.39 

43 

28.42 

48 

31 05 

34 

25.09 

39 

26.76 

44 

28.90 

49 

31.67 







50 

32.33 


































































































































TO FIND A TRUE MERIDIAN. 


123 


Second Method. 

Charles W. Helmick of Helena, Montana, furnishes 
the following method of determining a meridian by an 
observation on Polaris at any time. It obviates the 
use of astronomical time, and is extensively used by 
deputy mineral surveyors in Montana. 

1. Sight to Polaris and note the hour and minute of 
local mean solar time. 

2. Take from the ephemeris the time of elongation 
nearest the time of observation. The difference be¬ 
tween these times will be the hour angle east or west. 

3. Reduce this hour angle to degrees and minutes. 

4. Multiply the cosine of the resulting angle by the 
azimuth of Polaris for the latitude expressed in min¬ 
utes. The result will be the approximate azimuth of 
Polaris expressed in minutes. If the time piece gives 
local time within 2 minutes, the azimuth will rarely 
vary 1'. 




124 


A MANUAL OF LAND SURVEYING. 


To Find a True Meridian or Other Line by the 

Sun. 

We need for this work, first, the common engineer’s 
transit fitted with vertical arc or circle. A piece of 
shade or dark glass will be useful though it is not 
imperative. 

Second. The latitude of the place within a minute or 
two. This can readily be taken from a map. 

Third. A table of mean refraction containing less 
than half a hundred figures. 

Fourth. A table of the declination of the sun. This 
table is furnished gratis by many of our instrument 
makers. A good form is a vest-pocket pamphlet sent 
out by George N. Saegmuller, of Washington, D. C. A 
glance at the spherical triangle involved will show how 
easily the sun may be used to determine azimuth. 

In the figure, P represents the 
celestial pole of the earth, Z the 
zenith of the observer, and S the 
place of the sun at the time of 
observation. In the spherical tri¬ 
angle SPZ we know PZ, for it is 
the co-latitude of the place. ZS 
is the complement of AS, which 
latter is the altitude of the sun at 
the time of observation as measured by the transit. 
PS is the co-declination of the sun and is found from 
the table of declinations by subtracting the declina¬ 
tion from 90°, paying heed to the sign of the declina¬ 
tion. We then have the three sides of the triangle 
known and can readily compute the angle at Z by the 
formula 

sin IZ = / sin ( 4 S-p ) sin ( j S-s ) 

V sin P sin s 

in which p = co-altitude ; s = co-latitude, and z = co-de¬ 
clination and S = p + s+z. This angle gives the azi¬ 
muth of the sun at the instant of observation reckoned 
from the north point. The supplement of the angle 
would show its azimuth referred to the south point,, 
the usual origin of azimuth. 


Z 














TO FIND A TRUE MERIDIAN. 


125 


MOOtiONl Ai 


CH05S-Wi«£ 


Field Work .—About seven or eight o’clock in the 
morning or at five in the afternoon set the transit 
over a point commanding some prominent mark, as a 
church spire, flag pole or cupola to some building. Any 
time of the day will do except near noon, but the most 
favorable time except for uncertainties in refraction 
would be near sunrise or sunset. This appears, since 
we measure the altitude and from it compute the azi¬ 
muth at the time the sun is changing rapidly in alti¬ 
tude and slowly in azimuth. Hence only that part of 
an error in altitude will affect azimuth which is shown 
by the small change in azimuth compared with the ac¬ 
companying altitude change. Now bring the horizon¬ 
tal plates to zero and, being sure that 
the vertical circle reads zero for the col- 
limationline horizontal (or note its in¬ 
dex errors) and take a pointing at the 
sun. Abetter centering can be gained 
by bringing the horizontal cross wire 
very near the upper limb which gives a 
small segment to be bisected. Shown by the figure. 

Having with the lower horizontal slow motion gained 
center, with the vertical motion quickly move the hori¬ 
zontal wire to the upper limb of the sun. Read the 
vertical circle and loose the alidade plate and sight the 
selected terrestial mark and read the horizontal circle. 
Should we care not to depend upon a single determina¬ 
tion this operation may be repeated say four times, 
thus furnishing us not onl} T a mean value twice as ac¬ 
curate as a single observation but by the agreement ot 
the individual results among themselves we can judge 
much of their general reliability. By taking the sec¬ 
ond and fourth observations with the instrument in 



the reversed position —the full vertical circle permit¬ 
ting this— we eliminate both the error resulting from 
the horizontal axis of the instrument not being truly 
horizontal and the index error of the vertical circle. 

Sample observations made Aug. 24, 1895, and the 
method of computation, are given on the following 
page. 





126 


A MANUAL OP LAND SURVEYING. 


AZIMUTH OBSERVATIONS. 


Standard 

Time. 

Altitude of up¬ 
per limb. 

Horizontal cir¬ 
cle for center 
of sun. 

Horizontal cir¬ 
cle for Mark. 

9.06.45 

36° 01 l / 2 ' 

0.0 

141° 08J4* 


REMARKS. 

Transit over stone post set in courthouse yard. Mark is cu¬ 
pola on public schoolhouse. 

Latitude.44° 00' 

Temperature.80° 

Barometer.28.4 

Center of sun is azimuth and upper limb observed upon. 


Computation. 


Declination, corrected for dif. of time. = 11° 04' 

Co- “ or z . = 78° 56' 

Apparent altitude of upper limb . = 36° 01T 

Semi-diameter of sun. = 15V 

App. altitude of center of sun. = 35° 46' 

Refraction. = 1' 

True alt. of center of sun. = 35° 45' 

Co-altitude of sun, or p. — 54° 15' 

z = 78° 56' 
p = 54° 15' 
s = 46° 00’ 


S = 179° 11' 

|S = 89° 35V 


|S—p = 35° 201' log sin. = 9.762267 

IS—s = 43° 351' “ . = 9.838543 

p = 54° 15' “ A. C.= 0.090672 

s = 46° 00' “ A. C.= .143066 


19.834548 

1Z = 55° 44!'log sin. |Z. = 9.917274 

Z = 111° 29' reckoned from the north point. 

Azimuth of sun = 68° 31' reckoned from the south 
point. 

141° 081' — 68° 31' = 72° 37!' = azimuth of cupola 
on Schoolhouse. A transit at the station with hori¬ 
zontal circle reading 72° 371'and pointing at the cupola 
when returned to zero will point in the true meridian. 





























ABRIDGING FIELD NOTES. 


127 


The following method of abridging field notes is used 
by the land department of the United States. The 
plat of a township is lettered and numbered as shown 
in the diagram. Corners in the township boundary are 



referred to by letter; e. g., B or k. Interior section 
corners are referred to by the numbers of the sections; 
e. g., corner of 9, 10, 15, and 16. Interior quarter sec¬ 
tion corners are referred to by their position on the 
lines, e. g., K to W at 3 or B to G at 6. The descrip¬ 
tions of corners thus referred to are written out in the 
margins of the plats, while all other matter contained 
in the field notes is, as far as possible, marked on the 
plats themselves. The letters along the margin of the 
diagram are arranged the same as in Plate IV, Instruc¬ 
tions of 1894. A different arrangement has been used 
commencing in the upper left hand corner and passing 
around the plat in the opposite direction. 

































. 










: 


























' 






































































MANUAL 

—OF— 



LAND SURVEYING. 


By F. HODGMAN, M. S., C. E., 

Practical Surveyor and Civil Engineer. 


514 pages, printed on strong, light paper, and bound in morocco with flap. 

The Land Surveyor’s Best Pocket Companion. 

PRICE, - $2.50. 


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